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Proofgold Asset
asset id
a5d923df2f1c2aa58b81d07e6a6f96c66a504e4d3898eb8d458986431ff6f2d9
asset hash
4196cf46e5c079e261b3e9c6d6e1ceb054ca2abd138109b07201b2f0a835f878
bday / block
26895
tx
0304d..
preasset
doc published by
Pr5Zc..
Known
9560a..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x6
(
x1
x7
(
x1
x5
(
x1
x3
(
x1
x2
x9
)
)
)
)
)
)
Theorem
9c4a5..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x5
(
x1
x3
(
x1
x2
x7
)
)
)
)
)
)
(proof)
Theorem
03e79..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x5
(
x1
x3
(
x1
x2
x7
)
)
)
)
)
)
(proof)
Known
cbdc2..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
(
x1
x2
x9
)
)
)
)
)
)
Known
74985..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x3
(
x1
x6
(
x1
x4
(
x1
x7
(
x1
x5
(
x1
x2
x9
)
)
)
)
)
)
Theorem
431f5..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x5
(
x1
x7
(
x1
x3
x2
)
)
)
)
)
)
(proof)
Theorem
11e24..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x5
(
x1
x7
(
x1
x3
x2
)
)
)
)
)
)
(proof)
Known
c0c54..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
x8
)
)
)
)
)
=
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x2
x8
)
)
)
)
)
Theorem
0f5b5..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x5
(
x1
x7
(
x1
x2
x3
)
)
)
)
)
)
(proof)
Theorem
f279b..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x5
(
x1
x7
(
x1
x2
x3
)
)
)
)
)
)
(proof)
Known
93eac..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
x6
)
)
)
=
x1
x3
(
x1
x4
(
x1
x5
(
x1
x2
x6
)
)
)
Known
4961f..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x7
(
x1
x3
(
x1
x2
(
x1
x6
x9
)
)
)
)
)
)
Theorem
5892d..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x2
(
x1
x7
x5
)
)
)
)
)
)
(proof)
Theorem
92a24..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x2
(
x1
x7
x5
)
)
)
)
)
)
(proof)
Known
1029a..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x6
(
x1
x7
(
x1
x3
(
x1
x2
(
x1
x5
x9
)
)
)
)
)
)
Theorem
0341e..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x2
(
x1
x5
x7
)
)
)
)
)
)
(proof)
Theorem
9a22e..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x2
(
x1
x5
x7
)
)
)
)
)
)
(proof)
Known
b7d83..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x3
(
x1
x5
(
x1
x7
(
x1
x2
(
x1
x4
(
x1
x6
x9
)
)
)
)
)
)
Theorem
e87cd..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x5
(
x1
x7
x2
)
)
)
)
)
)
(proof)
Theorem
e5c96..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x5
(
x1
x7
x2
)
)
)
)
)
)
(proof)
Known
4dd1c..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x6
(
x1
x7
(
x1
x3
(
x1
x5
(
x1
x2
x9
)
)
)
)
)
)
Theorem
3e8c4..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x5
(
x1
x2
x7
)
)
)
)
)
)
(proof)
Theorem
23592..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x5
(
x1
x2
x7
)
)
)
)
)
)
(proof)
Known
0ad28..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x3
(
x1
x5
(
x1
x7
(
x1
x2
(
x1
x6
(
x1
x4
x9
)
)
)
)
)
)
Theorem
842a5..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x7
(
x1
x5
x2
)
)
)
)
)
)
(proof)
Theorem
d34a3..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x7
(
x1
x5
x2
)
)
)
)
)
)
(proof)
Known
20cf1..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x7
(
x1
x3
(
x1
x6
(
x1
x2
x9
)
)
)
)
)
)
Theorem
26097..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x7
(
x1
x2
x5
)
)
)
)
)
)
(proof)
Theorem
bd748..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x3
(
x1
x7
(
x1
x2
x5
)
)
)
)
)
)
(proof)
Known
5df2d..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x7
(
x1
x2
(
x1
x3
(
x1
x6
x9
)
)
)
)
)
)
Theorem
80158..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x3
(
x1
x7
x5
)
)
)
)
)
)
(proof)
Theorem
bdcb0..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x3
(
x1
x7
x5
)
)
)
)
)
)
(proof)
Known
a7494..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x6
(
x1
x7
(
x1
x2
(
x1
x3
(
x1
x5
x9
)
)
)
)
)
)
Theorem
bacae..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x3
(
x1
x5
x7
)
)
)
)
)
)
(proof)
Theorem
38c10..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x3
(
x1
x5
x7
)
)
)
)
)
)
(proof)
Theorem
d97cd..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x5
(
x1
x7
x3
)
)
)
)
)
)
(proof)
Theorem
186b7..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x5
(
x1
x7
x3
)
)
)
)
)
)
(proof)
Known
ec9a2..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x6
(
x1
x7
(
x1
x2
(
x1
x5
(
x1
x3
x9
)
)
)
)
)
)
Theorem
171d2..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x5
(
x1
x3
x7
)
)
)
)
)
)
(proof)
Theorem
451e7..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x5
(
x1
x3
x7
)
)
)
)
)
)
(proof)
Theorem
63a99..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x7
(
x1
x5
x3
)
)
)
)
)
)
(proof)
Theorem
39400..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x7
(
x1
x5
x3
)
)
)
)
)
)
(proof)
Known
3967d..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x7
(
x1
x2
(
x1
x6
(
x1
x3
x9
)
)
)
)
)
)
Theorem
9120e..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x7
(
x1
x3
x5
)
)
)
)
)
)
(proof)
Theorem
c96d7..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x6
(
x1
x8
(
x1
x2
(
x1
x7
(
x1
x3
x5
)
)
)
)
)
)
(proof)
Known
45f87..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x1
x2
(
x1
x3
(
x1
x4
x5
)
)
=
x1
x3
(
x1
x4
(
x1
x2
x5
)
)
Known
9751d..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x3
(
x1
x6
x9
)
)
)
)
)
)
Theorem
5b952..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x3
(
x1
x7
x6
)
)
)
)
)
)
(proof)
Theorem
b7f06..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x3
(
x1
x7
x6
)
)
)
)
)
)
(proof)
Theorem
e9c85..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x3
(
x1
x6
x7
)
)
)
)
)
)
(proof)
Theorem
e66a9..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x3
(
x1
x6
x7
)
)
)
)
)
)
(proof)
Known
04081..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x3
(
x1
x4
(
x1
x2
(
x1
x7
(
x1
x5
(
x1
x6
x9
)
)
)
)
)
)
Theorem
0ba29..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x6
(
x1
x7
x3
)
)
)
)
)
)
(proof)
Theorem
db1fe..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x6
(
x1
x7
x3
)
)
)
)
)
)
(proof)
Known
e6a7d..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x6
(
x1
x3
x9
)
)
)
)
)
)
Theorem
4223b..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x6
(
x1
x3
x7
)
)
)
)
)
)
(proof)
Theorem
58306..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x6
(
x1
x3
x7
)
)
)
)
)
)
(proof)
Known
ff57b..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x3
(
x1
x4
(
x1
x2
(
x1
x7
(
x1
x6
(
x1
x5
x9
)
)
)
)
)
)
Theorem
9d3c7..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x7
(
x1
x6
x3
)
)
)
)
)
)
(proof)
Theorem
3faba..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x7
(
x1
x6
x3
)
)
)
)
)
)
(proof)
Theorem
fba84..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x7
(
x1
x3
x6
)
)
)
)
)
)
(proof)
Theorem
c7424..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x7
(
x1
x3
x6
)
)
)
)
)
)
(proof)
Known
d39bf..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x3
(
x1
x7
x9
)
)
)
)
)
)
Theorem
80c54..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x3
(
x1
x8
x6
)
)
)
)
)
)
(proof)
Theorem
408e6..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x3
(
x1
x8
x6
)
)
)
)
)
)
(proof)
Theorem
3d6d8..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x3
(
x1
x6
x8
)
)
)
)
)
)
(proof)
Theorem
a8b31..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x3
(
x1
x6
x8
)
)
)
)
)
)
(proof)
Known
85826..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x3
(
x1
x4
(
x1
x2
(
x1
x6
(
x1
x5
(
x1
x7
x9
)
)
)
)
)
)
Theorem
25b1a..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x6
(
x1
x8
x3
)
)
)
)
)
)
(proof)
Theorem
650b3..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x6
(
x1
x8
x3
)
)
)
)
)
)
(proof)
Theorem
3762b..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x6
(
x1
x3
x8
)
)
)
)
)
)
(proof)
Theorem
8c1bb..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x6
(
x1
x3
x8
)
)
)
)
)
)
(proof)
Known
4c6f2..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x3
(
x1
x4
(
x1
x2
(
x1
x6
(
x1
x7
(
x1
x5
x9
)
)
)
)
)
)
Theorem
d228d..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x8
(
x1
x6
x3
)
)
)
)
)
)
(proof)
Theorem
a30e7..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x8
(
x1
x6
x3
)
)
)
)
)
)
(proof)
Known
33253..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x7
(
x1
x3
x9
)
)
)
)
)
)
Theorem
98152..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x8
(
x1
x3
x6
)
)
)
)
)
)
(proof)
Theorem
4efb1..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x8
(
x1
x3
x6
)
)
)
)
)
)
(proof)
Theorem
d8562..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x3
(
x1
x8
x7
)
)
)
)
)
)
(proof)
Theorem
0f3f8..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x3
(
x1
x8
x7
)
)
)
)
)
)
(proof)
Theorem
a620d..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x3
(
x1
x7
x8
)
)
)
)
)
)
(proof)
Theorem
d2689..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x3
(
x1
x7
x8
)
)
)
)
)
)
(proof)
Known
6533f..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x3
(
x1
x4
(
x1
x2
(
x1
x5
(
x1
x6
(
x1
x7
x9
)
)
)
)
)
)
Theorem
26c02..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x7
(
x1
x8
x3
)
)
)
)
)
)
(proof)
Theorem
8c749..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x7
(
x1
x8
x3
)
)
)
)
)
)
(proof)
Theorem
64568..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x7
(
x1
x3
x8
)
)
)
)
)
)
(proof)
Theorem
e0e38..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x7
(
x1
x3
x8
)
)
)
)
)
)
(proof)
Known
dc9ff..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x3
(
x1
x4
(
x1
x2
(
x1
x5
(
x1
x7
(
x1
x6
x9
)
)
)
)
)
)
Theorem
c82e1..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x8
(
x1
x7
x3
)
)
)
)
)
)
(proof)
Theorem
13abe..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x8
(
x1
x7
x3
)
)
)
)
)
)
(proof)
Theorem
e755b..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x8
(
x1
x3
x7
)
)
)
)
)
)
(proof)
Theorem
e43d8..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x6
(
x1
x8
(
x1
x3
x7
)
)
)
)
)
)
(proof)
Known
e4836..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x6
(
x1
x7
x9
)
)
)
)
)
)
Theorem
1980d..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x6
(
x1
x8
x7
)
)
)
)
)
)
(proof)
Theorem
f9c95..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x6
(
x1
x8
x7
)
)
)
)
)
)
(proof)
Theorem
cdeb5..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x6
(
x1
x7
x8
)
)
)
)
)
)
(proof)
Theorem
ee2d0..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x6
(
x1
x7
x8
)
)
)
)
)
)
(proof)
Theorem
a65ac..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x7
(
x1
x8
x6
)
)
)
)
)
)
(proof)
Theorem
ca7d5..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x7
(
x1
x8
x6
)
)
)
)
)
)
(proof)
Known
9a466..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x7
(
x1
x6
x9
)
)
)
)
)
)
Theorem
4974a..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x7
(
x1
x6
x8
)
)
)
)
)
)
(proof)
Theorem
37ad7..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x7
(
x1
x6
x8
)
)
)
)
)
)
(proof)
Theorem
063b2..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x8
(
x1
x7
x6
)
)
)
)
)
)
(proof)
Theorem
6337a..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x8
(
x1
x7
x6
)
)
)
)
)
)
(proof)
Theorem
bb4bc..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x8
(
x1
x6
x7
)
)
)
)
)
)
(proof)
Theorem
c65b5..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x8
(
x1
x6
x7
)
)
)
)
)
)
(proof)
Known
f7b65..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x2
(
x1
x6
x9
)
)
)
)
)
)
Theorem
8f436..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x2
(
x1
x7
x6
)
)
)
)
)
)
(proof)
Theorem
db8b7..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x2
(
x1
x7
x6
)
)
)
)
)
)
(proof)
Theorem
8c396..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x2
(
x1
x6
x7
)
)
)
)
)
)
(proof)
Theorem
b2fa0..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x2
(
x1
x6
x7
)
)
)
)
)
)
(proof)
Theorem
449bd..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x6
(
x1
x7
x2
)
)
)
)
)
)
(proof)
Theorem
eb839..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x6
(
x1
x7
x2
)
)
)
)
)
)
(proof)
Known
54d73..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x6
(
x1
x2
x9
)
)
)
)
)
)
Theorem
3d0d4..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x6
(
x1
x2
x7
)
)
)
)
)
)
(proof)
Theorem
06647..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x6
(
x1
x2
x7
)
)
)
)
)
)
(proof)
Theorem
28b51..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x7
(
x1
x6
x2
)
)
)
)
)
)
(proof)
Theorem
8821b..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x7
(
x1
x6
x2
)
)
)
)
)
)
(proof)
Theorem
de35f..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x7
(
x1
x2
x6
)
)
)
)
)
)
(proof)
Theorem
eea78..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x8
(
x1
x7
(
x1
x2
x6
)
)
)
)
)
)
(proof)
Known
5021d..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x6
(
x1
x2
(
x1
x7
x9
)
)
)
)
)
)
Theorem
1dfc0..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x2
(
x1
x8
x6
)
)
)
)
)
)
(proof)
Theorem
aea70..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x2
(
x1
x8
x6
)
)
)
)
)
)
(proof)
Theorem
6a0ce..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x2
(
x1
x6
x8
)
)
)
)
)
)
(proof)
Theorem
ddc19..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x2
(
x1
x6
x8
)
)
)
)
)
)
(proof)
Theorem
a3005..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x6
(
x1
x8
x2
)
)
)
)
)
)
(proof)
Theorem
9f9cd..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x6
(
x1
x8
x2
)
)
)
)
)
)
(proof)
Theorem
78d6a..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x6
(
x1
x2
x8
)
)
)
)
)
)
(proof)
Theorem
4b324..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x6
(
x1
x2
x8
)
)
)
)
)
)
(proof)
Theorem
e4fc8..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x8
(
x1
x6
x2
)
)
)
)
)
)
(proof)
Theorem
d2f4d..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x8
(
x1
x6
x2
)
)
)
)
)
)
(proof)
Known
73e83..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x8
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x6
(
x1
x7
(
x1
x2
x9
)
)
)
)
)
)
Theorem
bb358..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x8
(
x1
x2
x6
)
)
)
)
)
)
(proof)
Theorem
445a5..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x7
(
x1
x8
(
x1
x2
x6
)
)
)
)
)
)
(proof)
Theorem
f3b0f..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x6
(
x1
x2
(
x1
x8
x7
)
)
)
)
)
)
(proof)
Theorem
78c25..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x6
(
x1
x2
(
x1
x8
x7
)
)
)
)
)
)
(proof)
Theorem
5581b..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
x3
(
x1
x2
x4
)
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x6
(
x1
x2
(
x1
x7
x8
)
)
)
)
)
)
(proof)
Theorem
992d5..
:
∀ x0 :
ι → ο
.
∀ x1 :
ι →
ι → ι
.
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x0
(
x1
x2
x3
)
)
⟶
(
∀ x2 x3 x4 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x1
x2
(
x1
x3
x4
)
=
x1
(
x1
x2
x3
)
x4
)
⟶
(
∀ x2 x3 .
x0
x2
⟶
x0
x3
⟶
x1
x2
x3
=
x1
x3
x2
)
⟶
∀ x2 x3 x4 x5 x6 x7 x8 x9 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x0
x8
⟶
x0
x9
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x8
x9
)
)
)
)
)
)
=
x1
x9
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x6
(
x1
x2
(
x1
x7
x8
)
)
)
)
)
)
(proof)