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Proofgold Asset
asset id
43a8550de5015f071eafdf9ba09e6ac6b6f42e37a74a827109c94b7444efb27d
asset hash
dcb7b49a326770a70c581960bd96e02a8e47e471110a0aada0b17dd42ae77666
bday / block
24613
tx
c645c..
preasset
doc published by
Pr5Zc..
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
)
)
Theorem
c2dad..
:
∀ 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
x5
(
x1
x2
(
x1
x4
(
x1
x3
x6
)
)
)
(proof)
Theorem
ac781..
:
∀ 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
x5
(
x1
x3
(
x1
x4
(
x1
x2
x6
)
)
)
(proof)
Theorem
b2677..
:
∀ 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
x5
(
x1
x3
(
x1
x2
(
x1
x4
x6
)
)
)
(proof)
Known
8c2ea..
:
∀ 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
x4
(
x1
x3
(
x1
x2
x5
)
)
Theorem
12698..
:
∀ 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
x5
(
x1
x4
(
x1
x3
(
x1
x2
x6
)
)
)
(proof)
Theorem
c09e5..
:
∀ 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
x5
(
x1
x4
(
x1
x2
(
x1
x3
x6
)
)
)
(proof)
Theorem
f7707..
:
∀ 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
x4
(
x1
x2
(
x1
x5
(
x1
x3
x6
)
)
)
(proof)
Theorem
2bf06..
:
∀ 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
x2
(
x1
x5
(
x1
x4
x6
)
)
)
(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
)
)
)
Theorem
afb35..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x2
(
x1
x5
(
x1
x3
(
x1
x4
x7
)
)
)
)
(proof)
Theorem
75ac7..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x2
(
x1
x3
(
x1
x5
(
x1
x4
x7
)
)
)
)
(proof)
Theorem
bd148..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x3
(
x1
x5
(
x1
x2
(
x1
x4
x7
)
)
)
)
(proof)
Theorem
a4b60..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x3
(
x1
x4
(
x1
x2
(
x1
x5
x7
)
)
)
)
(proof)
Theorem
303f8..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x3
(
x1
x2
(
x1
x4
(
x1
x5
x7
)
)
)
)
(proof)
Theorem
cbff5..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x3
(
x1
x2
(
x1
x5
(
x1
x4
x7
)
)
)
)
(proof)
Known
0d20b..
:
∀ 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
x4
(
x1
x5
(
x1
x2
(
x1
x3
x6
)
)
)
Theorem
0f4fc..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
x7
)
)
)
)
(proof)
Theorem
58437..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x4
(
x1
x3
(
x1
x2
(
x1
x5
x7
)
)
)
)
(proof)
Theorem
7cae8..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x4
(
x1
x2
(
x1
x3
(
x1
x5
x7
)
)
)
)
(proof)
Theorem
a445d..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x4
(
x1
x2
(
x1
x5
(
x1
x3
x7
)
)
)
)
(proof)
Theorem
8c4b6..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x5
(
x1
x3
(
x1
x2
(
x1
x4
x7
)
)
)
)
(proof)
Theorem
76f9e..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x4
x7
)
)
)
)
(proof)
Theorem
4d854..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x6
(
x1
x5
(
x1
x2
(
x1
x4
(
x1
x3
x7
)
)
)
)
(proof)
Theorem
d5477..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x5
(
x1
x2
(
x1
x6
(
x1
x3
(
x1
x4
x7
)
)
)
)
(proof)
Theorem
9007e..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x5
(
x1
x2
(
x1
x4
(
x1
x6
(
x1
x3
x7
)
)
)
)
(proof)
Theorem
17962..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x5
(
x1
x2
(
x1
x3
(
x1
x6
(
x1
x4
x7
)
)
)
)
(proof)
Theorem
956b1..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x5
(
x1
x3
(
x1
x2
(
x1
x6
(
x1
x4
x7
)
)
)
)
(proof)
Theorem
cd0f4..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x4
(
x1
x2
(
x1
x6
(
x1
x3
(
x1
x5
x7
)
)
)
)
(proof)
Theorem
92a54..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x4
(
x1
x2
(
x1
x6
(
x1
x5
(
x1
x3
x7
)
)
)
)
(proof)
Theorem
bbbe4..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x3
(
x1
x2
(
x1
x6
(
x1
x4
(
x1
x5
x7
)
)
)
)
(proof)
Theorem
4a5b9..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x3
(
x1
x2
(
x1
x6
(
x1
x5
(
x1
x4
x7
)
)
)
)
(proof)
Known
f87dc..
:
∀ 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 .
x0
x2
⟶
x0
x3
⟶
x0
x4
⟶
x0
x5
⟶
x0
x6
⟶
x0
x7
⟶
x1
x2
(
x1
x3
(
x1
x4
(
x1
x5
(
x1
x6
x7
)
)
)
)
=
x1
x4
(
x1
x5
(
x1
x6
(
x1
x2
(
x1
x3
x7
)
)
)
)
Theorem
f3668..
:
∀ 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
x6
(
x1
x3
(
x1
x7
(
x1
x2
(
x1
x4
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
3abe0..
:
∀ 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
x6
(
x1
x3
(
x1
x4
(
x1
x2
(
x1
x7
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
fa98b..
:
∀ 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
x6
(
x1
x3
(
x1
x2
(
x1
x4
(
x1
x7
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
9ae99..
:
∀ 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
x6
(
x1
x3
(
x1
x2
(
x1
x7
(
x1
x4
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
caca7..
:
∀ 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
x6
(
x1
x4
(
x1
x7
(
x1
x2
(
x1
x5
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
f285b..
:
∀ 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
x6
(
x1
x4
(
x1
x7
(
x1
x2
(
x1
x3
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
b35e9..
:
∀ 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
x6
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
11442..
:
∀ 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
x6
(
x1
x4
(
x1
x3
(
x1
x2
(
x1
x7
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
9b1af..
:
∀ 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
x6
(
x1
x4
(
x1
x2
(
x1
x3
(
x1
x7
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
4dc29..
:
∀ 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
x6
(
x1
x4
(
x1
x2
(
x1
x5
(
x1
x7
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
bf22a..
:
∀ 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
x6
(
x1
x4
(
x1
x2
(
x1
x7
(
x1
x5
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
a283f..
:
∀ 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
x6
(
x1
x4
(
x1
x2
(
x1
x7
(
x1
x3
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
6bf93..
:
∀ 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
x6
(
x1
x5
(
x1
x7
(
x1
x2
(
x1
x4
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
e9e2e..
:
∀ 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
x6
(
x1
x5
(
x1
x2
(
x1
x4
(
x1
x7
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
9492b..
:
∀ 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
x6
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x4
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
3d147..
:
∀ 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
x6
(
x1
x7
(
x1
x5
(
x1
x2
(
x1
x4
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
0ab51..
:
∀ 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
x6
(
x1
x7
(
x1
x4
(
x1
x2
(
x1
x5
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
7230f..
:
∀ 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
x6
(
x1
x7
(
x1
x4
(
x1
x2
(
x1
x3
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
ba174..
:
∀ 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
x6
(
x1
x7
(
x1
x3
(
x1
x2
(
x1
x4
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
e1c79..
:
∀ 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
x6
(
x1
x7
(
x1
x2
(
x1
x4
(
x1
x5
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
e1ce7..
:
∀ 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
x6
(
x1
x7
(
x1
x2
(
x1
x4
(
x1
x3
(
x1
x5
x8
)
)
)
)
)
(proof)
Theorem
6e77e..
:
∀ 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
x6
(
x1
x7
(
x1
x2
(
x1
x5
(
x1
x4
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
59b72..
:
∀ 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
x5
(
x1
x2
(
x1
x4
(
x1
x3
(
x1
x7
(
x1
x6
x8
)
)
)
)
)
(proof)
Theorem
151c0..
:
∀ 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
x5
(
x1
x2
(
x1
x4
(
x1
x6
(
x1
x7
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
f27e3..
:
∀ 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
x5
(
x1
x2
(
x1
x4
(
x1
x7
(
x1
x6
(
x1
x3
x8
)
)
)
)
)
(proof)
Theorem
cb824..
:
∀ 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
x5
(
x1
x2
(
x1
x4
(
x1
x7
(
x1
x3
(
x1
x6
x8
)
)
)
)
)
(proof)
Theorem
1597f..
:
∀ 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
x5
(
x1
x2
(
x1
x3
(
x1
x7
(
x1
x4
(
x1
x6
x8
)
)
)
)
)
(proof)
Known
209c8..
:
∀ 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
x5
(
x1
x6
(
x1
x7
(
x1
x2
(
x1
x3
(
x1
x4
x8
)
)
)
)
)
Theorem
48d5c..
:
∀ 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
x6
(
x1
x3
(
x1
x2
(
x1
x7
(
x1
x5
(
x1
x8
(
x1
x4
x9
)
)
)
)
)
)
(proof)
Theorem
717bc..
:
∀ 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
x6
(
x1
x3
(
x1
x2
(
x1
x7
(
x1
x4
(
x1
x8
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
483a4..
:
∀ 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
x6
(
x1
x3
(
x1
x2
(
x1
x8
(
x1
x7
(
x1
x4
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
4a7f4..
:
∀ 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
x6
(
x1
x3
(
x1
x2
(
x1
x8
(
x1
x7
(
x1
x5
(
x1
x4
x9
)
)
)
)
)
)
(proof)
Theorem
a4e3d..
:
∀ 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
x6
(
x1
x3
(
x1
x2
(
x1
x8
(
x1
x5
(
x1
x4
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
171cf..
:
∀ 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
x6
(
x1
x3
(
x1
x2
(
x1
x8
(
x1
x5
(
x1
x7
(
x1
x4
x9
)
)
)
)
)
)
(proof)
Theorem
7b320..
:
∀ 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
x6
(
x1
x3
(
x1
x2
(
x1
x8
(
x1
x4
(
x1
x5
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
05b11..
:
∀ 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
x6
(
x1
x3
(
x1
x2
(
x1
x8
(
x1
x4
(
x1
x7
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
28361..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x2
(
x1
x7
(
x1
x3
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
ef279..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x2
(
x1
x7
(
x1
x5
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
541a3..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x2
(
x1
x5
(
x1
x3
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
f897d..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x2
(
x1
x5
(
x1
x7
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
0f9ee..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x2
(
x1
x3
(
x1
x5
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
ba093..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x2
(
x1
x3
(
x1
x7
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
4ec2d..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x3
(
x1
x2
(
x1
x5
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
79188..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x3
(
x1
x2
(
x1
x7
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
db017..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
13833..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
3f7d3..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x7
(
x1
x2
(
x1
x3
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
4fb90..
:
∀ 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
x6
(
x1
x4
(
x1
x8
(
x1
x7
(
x1
x2
(
x1
x5
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
ed028..
:
∀ 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
x6
(
x1
x4
(
x1
x7
(
x1
x2
(
x1
x8
(
x1
x3
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
53c1d..
:
∀ 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
x6
(
x1
x4
(
x1
x7
(
x1
x2
(
x1
x8
(
x1
x5
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
88ebd..
:
∀ 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
x6
(
x1
x4
(
x1
x7
(
x1
x2
(
x1
x5
(
x1
x8
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
f70fa..
:
∀ 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
x6
(
x1
x4
(
x1
x7
(
x1
x2
(
x1
x3
(
x1
x8
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
0a7c2..
:
∀ 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
x6
(
x1
x4
(
x1
x7
(
x1
x3
(
x1
x2
(
x1
x8
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
d3b84..
:
∀ 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
x6
(
x1
x4
(
x1
x7
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
63d4c..
:
∀ 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
x6
(
x1
x4
(
x1
x7
(
x1
x8
(
x1
x2
(
x1
x3
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
f687d..
:
∀ 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
x6
(
x1
x4
(
x1
x7
(
x1
x8
(
x1
x2
(
x1
x5
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
a508b..
:
∀ 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
x6
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x3
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
20873..
:
∀ 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
x6
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x7
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Known
3a13f..
:
∀ 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
x4
(
x1
x5
(
x1
x6
(
x1
x7
(
x1
x2
(
x1
x3
x8
)
)
)
)
)
Theorem
51e9c..
:
∀ 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
x6
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x7
(
x1
x8
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
843b0..
:
∀ 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
x6
(
x1
x4
(
x1
x5
(
x1
x2
(
x1
x3
(
x1
x8
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
7eb11..
:
∀ 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
x6
(
x1
x4
(
x1
x5
(
x1
x3
(
x1
x2
(
x1
x8
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
46ed9..
:
∀ 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
x6
(
x1
x4
(
x1
x5
(
x1
x7
(
x1
x2
(
x1
x8
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
520d3..
:
∀ 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
x6
(
x1
x4
(
x1
x5
(
x1
x8
(
x1
x2
(
x1
x3
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
fb3a1..
:
∀ 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
x6
(
x1
x4
(
x1
x5
(
x1
x8
(
x1
x2
(
x1
x7
(
x1
x3
x9
)
)
)
)
)
)
(proof)
Theorem
96dd3..
:
∀ 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
x6
(
x1
x4
(
x1
x3
(
x1
x2
(
x1
x8
(
x1
x5
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
16b8f..
:
∀ 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
x6
(
x1
x4
(
x1
x3
(
x1
x2
(
x1
x8
(
x1
x7
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
64f44..
:
∀ 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
x6
(
x1
x4
(
x1
x3
(
x1
x2
(
x1
x7
(
x1
x8
(
x1
x5
x9
)
)
)
)
)
)
(proof)
Theorem
da5be..
:
∀ 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
x6
(
x1
x4
(
x1
x3
(
x1
x2
(
x1
x5
(
x1
x8
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
a3312..
:
∀ 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
x6
(
x1
x4
(
x1
x3
(
x1
x5
(
x1
x2
(
x1
x8
(
x1
x7
x9
)
)
)
)
)
)
(proof)
Theorem
7d710..
:
∀ 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
x6
(
x1
x4
(
x1
x3
(
x1
x7
(
x1
x2
(
x1
x8
(
x1
x5
x9
)
)
)
)
)
)
(proof)