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address
PUfssg5E7wSdDPVJeiGj7T5ry1BocYWpE2i
total
0
mg
-
conjpub
-
current assets
173a4..
/
5e2af..
bday:
2718
doc published by
PrGxv..
Definition
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Theorem
exandE_i
:
∀ x0 x1 :
ι → ο
.
(
∀ x2 : ο .
(
∀ x3 .
and
(
x0
x3
)
(
x1
x3
)
⟶
x2
)
⟶
x2
)
⟶
∀ x2 : ο .
(
∀ x3 .
x0
x3
⟶
x1
x3
⟶
x2
)
⟶
x2
(proof)
Definition
False
:=
∀ x0 : ο .
x0
Definition
not
:=
λ x0 : ο .
x0
⟶
False
Definition
If_Vo4
:=
λ x0 : ο .
λ x1 x2 :
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
λ x3 :
(
(
ι → ο
)
→ ο
)
→ ο
.
and
(
x0
⟶
x1
x3
)
(
not
x0
⟶
x2
x3
)
Known
prop_ext_2
:
∀ x0 x1 : ο .
(
x0
⟶
x1
)
⟶
(
x1
⟶
x0
)
⟶
x0
=
x1
Known
andEL
:
∀ x0 x1 : ο .
and
x0
x1
⟶
x0
Known
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Known
FalseE
:
False
⟶
∀ x0 : ο .
x0
Known
notE
:
∀ x0 : ο .
not
x0
⟶
x0
⟶
False
Theorem
If_Vo4_1
:
∀ x0 : ο .
∀ x1 x2 :
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
x0
⟶
If_Vo4
x0
x1
x2
=
x1
(proof)
Known
andER
:
∀ x0 x1 : ο .
and
x0
x1
⟶
x1
Theorem
If_Vo4_0
:
∀ x0 : ο .
∀ x1 x2 :
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
not
x0
⟶
If_Vo4
x0
x1
x2
=
x2
(proof)
Definition
Descr_Vo4
:=
λ x0 :
(
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→ ο
.
λ x1 :
(
(
ι → ο
)
→ ο
)
→ ο
.
∀ x2 :
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
x0
x2
⟶
x2
x1
Theorem
Descr_Vo4_prop
:
∀ x0 :
(
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x1 : ο .
(
∀ x2 :
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
x0
x2
⟶
x1
)
⟶
x1
)
⟶
(
∀ x1 x2 :
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
x0
x1
⟶
x0
x2
⟶
x1
=
x2
)
⟶
x0
(
Descr_Vo4
x0
)
(proof)
Definition
9eb9b..
:=
λ x0 :
ι →
(
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
λ x1 .
λ x2 :
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
∀ x3 :
ι →
(
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x4 .
∀ x5 :
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x6 .
prim1
x6
x4
⟶
x3
x6
(
x5
x6
)
)
⟶
x3
x4
(
x0
x4
x5
)
)
⟶
x3
x1
x2
Definition
In_rec_Vo4
:=
λ x0 :
ι →
(
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
λ x1 .
Descr_Vo4
(
9eb9b..
x0
x1
)
Theorem
71e94..
:
∀ x0 :
ι →
(
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
∀ x1 .
∀ x2 :
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x3 .
prim1
x3
x1
⟶
9eb9b..
x0
x3
(
x2
x3
)
)
⟶
9eb9b..
x0
x1
(
x0
x1
x2
)
(proof)
Theorem
1c22d..
:
∀ x0 :
ι →
(
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
∀ x1 .
∀ x2 :
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
9eb9b..
x0
x1
x2
⟶
∀ x3 : ο .
(
∀ x4 :
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
and
(
∀ x5 .
prim1
x5
x1
⟶
9eb9b..
x0
x5
(
x4
x5
)
)
(
x2
=
x0
x1
x4
)
⟶
x3
)
⟶
x3
(proof)
Known
In_ind
:
∀ x0 :
ι → ο
.
(
∀ x1 .
(
∀ x2 .
prim1
x2
x1
⟶
x0
x2
)
⟶
x0
x1
)
⟶
∀ x1 .
x0
x1
Theorem
0f420..
:
∀ x0 :
ι →
(
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x1 .
∀ x2 x3 :
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x4 .
prim1
x4
x1
⟶
x2
x4
=
x3
x4
)
⟶
x0
x1
x2
=
x0
x1
x3
)
⟶
∀ x1 .
∀ x2 x3 :
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
9eb9b..
x0
x1
x2
⟶
9eb9b..
x0
x1
x3
⟶
x2
=
x3
(proof)
Theorem
f8b5f..
:
∀ x0 :
ι →
(
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x1 .
∀ x2 x3 :
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x4 .
prim1
x4
x1
⟶
x2
x4
=
x3
x4
)
⟶
x0
x1
x2
=
x0
x1
x3
)
⟶
∀ x1 .
9eb9b..
x0
x1
(
In_rec_Vo4
x0
x1
)
(proof)
Theorem
eb697..
:
∀ x0 :
ι →
(
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x1 .
∀ x2 x3 :
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x4 .
prim1
x4
x1
⟶
x2
x4
=
x3
x4
)
⟶
x0
x1
x2
=
x0
x1
x3
)
⟶
∀ x1 .
9eb9b..
x0
x1
(
x0
x1
(
In_rec_Vo4
x0
)
)
(proof)
Theorem
In_rec_Vo4_eq
:
∀ x0 :
ι →
(
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
)
→
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x1 .
∀ x2 x3 :
ι →
(
(
(
ι → ο
)
→ ο
)
→ ο
)
→ ο
.
(
∀ x4 .
prim1
x4
x1
⟶
x2
x4
=
x3
x4
)
⟶
x0
x1
x2
=
x0
x1
x3
)
⟶
∀ x1 .
In_rec_Vo4
x0
x1
=
x0
x1
(
In_rec_Vo4
x0
)
(proof)
Definition
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Known
xm
:
∀ x0 : ο .
or
x0
(
not
x0
)
Known
orIR
:
∀ x0 x1 : ο .
x1
⟶
or
x0
x1
Known
orIL
:
∀ x0 x1 : ο .
x0
⟶
or
x0
x1
Theorem
not_and_or_demorgan
:
∀ x0 x1 : ο .
not
(
and
x0
x1
)
⟶
or
(
not
x0
)
(
not
x1
)
(proof)
Theorem
eq_imp_or
:
(
λ x1 x2 : ο .
x1
⟶
x2
)
=
λ x1 : ο .
or
(
not
x1
)
(proof)
Definition
Subq
:=
λ x0 x1 .
∀ x2 .
prim1
x2
x0
⟶
prim1
x2
x1
Definition
nIn
:=
λ x0 x1 .
not
(
prim1
x0
x1
)
Theorem
Subq_contra
:
∀ x0 x1 x2 .
Subq
x0
x1
⟶
nIn
x2
x1
⟶
nIn
x2
x0
(proof)
Param
4a7ef..
:
ι
Known
set_ext
:
∀ x0 x1 .
Subq
x0
x1
⟶
Subq
x1
x0
⟶
x0
=
x1
Known
e8942..
:
∀ x0 .
Subq
4a7ef..
x0
Theorem
3f849..
:
∀ x0 .
Subq
x0
4a7ef..
⟶
x0
=
4a7ef..
(proof)
Theorem
3c674..
:
∀ x0 .
(
∀ x1 .
nIn
x1
x0
)
⟶
x0
=
4a7ef..
(proof)
Known
UnionE
:
∀ x0 x1 .
prim1
x1
(
prim3
x0
)
⟶
∀ x2 : ο .
(
∀ x3 .
and
(
prim1
x1
x3
)
(
prim1
x3
x0
)
⟶
x2
)
⟶
x2
Known
dcd83..
:
∀ x0 .
nIn
x0
4a7ef..
Theorem
b3f0d..
:
prim3
4a7ef..
=
4a7ef..
(proof)
Param
e5b72..
:
ι
→
ι
Known
b21da..
:
∀ x0 x1 .
prim1
x1
(
e5b72..
x0
)
⟶
Subq
x1
x0
Theorem
81acb..
:
∀ x0 .
Subq
(
prim3
(
e5b72..
x0
)
)
x0
(proof)
Param
94f9e..
:
ι
→
(
ι
→
ι
) →
ι
Known
8c6f6..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
prim1
x2
(
94f9e..
x0
x1
)
⟶
∀ x3 : ο .
(
∀ x4 .
prim1
x4
x0
⟶
x2
=
x1
x4
⟶
x3
)
⟶
x3
Theorem
d4dbc..
:
∀ x0 :
ι → ι
.
94f9e..
4a7ef..
x0
=
4a7ef..
(proof)
Known
696c0..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
prim1
x2
x0
⟶
prim1
(
x1
x2
)
(
94f9e..
x0
x1
)
Theorem
02c03..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
(
∀ x3 .
prim1
x3
x0
⟶
x1
x3
=
x2
x3
)
⟶
Subq
(
94f9e..
x0
x1
)
(
94f9e..
x0
x2
)
(proof)
Theorem
aff96..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
(
∀ x3 .
prim1
x3
x0
⟶
x1
x3
=
x2
x3
)
⟶
94f9e..
x0
x1
=
94f9e..
x0
x2
(proof)
Known
2532b..
:
∀ x0 x1 x2 .
prim1
x0
(
prim2
x1
x2
)
⟶
or
(
x0
=
x1
)
(
x0
=
x2
)
Known
5a932..
:
∀ x0 x1 .
prim1
x1
(
prim2
x0
x1
)
Known
67787..
:
∀ x0 x1 .
prim1
x0
(
prim2
x0
x1
)
Theorem
38ffc..
:
∀ x0 x1 .
Subq
(
prim2
x0
x1
)
(
prim2
x1
x0
)
(proof)
Theorem
7cb28..
:
∀ x0 x1 .
prim2
x0
x1
=
prim2
x1
x0
(proof)
Param
0ac37..
:
ι
→
ι
→
ι
Param
91630..
:
ι
→
ι
Definition
15418..
:=
λ x0 x1 .
0ac37..
x0
(
91630..
x1
)
Known
e7a4c..
:
∀ x0 .
prim1
x0
(
91630..
x0
)
Known
fead7..
:
∀ x0 x1 .
prim1
x1
(
91630..
x0
)
⟶
x1
=
x0
Known
c5d9a..
:
∀ x0 .
prim1
4a7ef..
(
e5b72..
x0
)
Theorem
640be..
:
e5b72..
4a7ef..
=
91630..
4a7ef..
(proof)
Theorem
2b8c2..
:
∀ x0 :
ι → ι
.
∀ x1 x2 .
94f9e..
(
prim2
x1
x2
)
x0
=
prim2
(
x0
x1
)
(
x0
x2
)
(proof)
Theorem
535ee..
:
∀ x0 :
ι → ι
.
∀ x1 .
94f9e..
(
91630..
x1
)
x0
=
91630..
(
x0
x1
)
(proof)
Known
6acac..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
prim1
x2
(
94f9e..
x0
x1
)
⟶
∀ x3 : ο .
(
∀ x4 .
and
(
prim1
x4
x0
)
(
x2
=
x1
x4
)
⟶
x3
)
⟶
x3
Theorem
02c03..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
(
∀ x3 .
prim1
x3
x0
⟶
x1
x3
=
x2
x3
)
⟶
Subq
(
94f9e..
x0
x1
)
(
94f9e..
x0
x2
)
(proof)
Theorem
aff96..
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
(
∀ x3 .
prim1
x3
x0
⟶
x1
x3
=
x2
x3
)
⟶
94f9e..
x0
x1
=
94f9e..
x0
x2
(proof)
Definition
a842e..
:=
λ x0 .
λ x1 :
ι → ι
.
prim3
(
94f9e..
x0
x1
)
Known
UnionI
:
∀ x0 x1 x2 .
prim1
x1
x2
⟶
prim1
x2
x0
⟶
prim1
x1
(
prim3
x0
)
Theorem
2236b..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 x3 .
prim1
x2
x0
⟶
prim1
x3
(
x1
x2
)
⟶
prim1
x3
(
a842e..
x0
x1
)
(proof)
Known
UnionE_impred
:
∀ x0 x1 .
prim1
x1
(
prim3
x0
)
⟶
∀ x2 : ο .
(
∀ x3 .
prim1
x1
x3
⟶
prim1
x3
x0
⟶
x2
)
⟶
x2
Theorem
042d7..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
prim1
x2
(
a842e..
x0
x1
)
⟶
∀ x3 : ο .
(
∀ x4 .
and
(
prim1
x4
x0
)
(
prim1
x2
(
x1
x4
)
)
⟶
x3
)
⟶
x3
(proof)
Theorem
9b5af..
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
prim1
x2
(
a842e..
x0
x1
)
⟶
∀ x3 : ο .
(
∀ x4 .
prim1
x4
x0
⟶
prim1
x2
(
x1
x4
)
⟶
x3
)
⟶
x3
(proof)
Theorem
0a29e..
:
∀ x0 .
prim3
x0
=
a842e..
x0
(
λ x2 .
x2
)
(proof)
Theorem
06b06..
:
∀ x0 .
∀ x1 :
ι → ι
.
94f9e..
x0
x1
=
a842e..
x0
(
λ x3 .
91630..
(
x1
x3
)
)
(proof)
Theorem
f8e95..
:
∀ x0 .
or
(
x0
=
4a7ef..
)
(
∀ x1 : ο .
(
∀ x2 .
prim1
x2
x0
⟶
x1
)
⟶
x1
)
(proof)
Known
62c05..
:
∀ x0 .
prim1
x0
(
e5b72..
x0
)
Theorem
04a44..
:
∀ x0 .
e5b72..
(
91630..
x0
)
=
prim2
4a7ef..
(
91630..
x0
)
(proof)
Theorem
e8b5a..
:
e5b72..
(
91630..
4a7ef..
)
=
prim2
4a7ef..
(
91630..
4a7ef..
)
(proof)
Param
1216a..
:
ι
→
(
ι
→
ο
) →
ι
Known
d0a1f..
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
prim1
x2
(
1216a..
x0
x1
)
⟶
prim1
x2
x0
Theorem
572ea..
:
∀ x0 .
∀ x1 :
ι → ο
.
Subq
(
1216a..
x0
x1
)
x0
(proof)
Known
3daee..
:
∀ x0 x1 .
Subq
x1
x0
⟶
prim1
x1
(
e5b72..
x0
)
Theorem
d129e..
:
∀ x0 .
∀ x1 :
ι → ο
.
prim1
(
1216a..
x0
x1
)
(
e5b72..
x0
)
(proof)
Known
edd11..
:
∀ x0 x1 x2 .
prim1
x2
(
0ac37..
x0
x1
)
⟶
or
(
prim1
x2
x0
)
(
prim1
x2
x1
)
Known
da962..
:
∀ x0 x1 x2 .
prim1
x2
x0
⟶
prim1
x2
(
0ac37..
x0
x1
)
Known
287ca..
:
∀ x0 x1 x2 .
prim1
x2
x1
⟶
prim1
x2
(
0ac37..
x0
x1
)
Theorem
cf5a9..
:
∀ x0 x1 x2 .
0ac37..
x0
(
0ac37..
x1
x2
)
=
0ac37..
(
0ac37..
x0
x1
)
x2
(proof)
Theorem
64b03..
:
∀ x0 x1 .
Subq
(
0ac37..
x0
x1
)
(
0ac37..
x1
x0
)
(proof)
Theorem
47e6b..
:
∀ x0 x1 .
0ac37..
x0
x1
=
0ac37..
x1
x0
(proof)
Theorem
019ee..
:
∀ x0 .
0ac37..
4a7ef..
x0
=
x0
(proof)
Theorem
4d5c9..
:
∀ x0 .
0ac37..
x0
4a7ef..
=
x0
(proof)
Theorem
c20d9..
:
∀ x0 .
0ac37..
x0
x0
=
x0
(proof)
Theorem
9c551..
:
∀ x0 x1 .
Subq
x0
(
0ac37..
x0
x1
)
(proof)
Theorem
cde9c..
:
∀ x0 x1 .
Subq
x1
(
0ac37..
x0
x1
)
(proof)
Theorem
011e9..
:
∀ x0 x1 x2 .
Subq
x0
x2
⟶
Subq
x1
x2
⟶
Subq
(
0ac37..
x0
x1
)
x2
(proof)
Known
Subq_ref
:
∀ x0 .
Subq
x0
x0
Theorem
02255..
:
∀ x0 x1 .
Subq
x0
x1
=
(
0ac37..
x0
x1
=
x1
)
(proof)
Theorem
82f52..
:
∀ x0 x1 x2 .
nIn
x2
x0
⟶
nIn
x2
x1
⟶
nIn
x2
(
0ac37..
x0
x1
)
(proof)
Theorem
7b5e9..
:
∀ x0 x1 x2 .
nIn
x2
(
0ac37..
x0
x1
)
⟶
and
(
nIn
x2
x0
)
(
nIn
x2
x1
)
(proof)
Param
d3786..
:
ι
→
ι
→
ι
Known
695d8..
:
∀ x0 x1 x2 .
prim1
x2
(
d3786..
x0
x1
)
⟶
prim1
x2
x0
Theorem
fb39c..
:
∀ x0 x1 .
Subq
(
d3786..
x0
x1
)
x0
(proof)
Known
a5fe0..
:
∀ x0 x1 x2 .
prim1
x2
(
d3786..
x0
x1
)
⟶
prim1
x2
x1
Theorem
93cc4..
:
∀ x0 x1 .
Subq
(
d3786..
x0
x1
)
x1
(proof)
Known
2d07f..
:
∀ x0 x1 x2 .
prim1
x2
x0
⟶
prim1
x2
x1
⟶
prim1
x2
(
d3786..
x0
x1
)
Theorem
bd118..
:
∀ x0 x1 .
Subq
x0
x1
⟶
d3786..
x0
x1
=
x0
(proof)
Theorem
fd472..
:
∀ x0 x1 x2 .
Subq
x2
x0
⟶
Subq
x2
x1
⟶
Subq
x2
(
d3786..
x0
x1
)
(proof)
Known
Subq_tra
:
∀ x0 x1 x2 .
Subq
x0
x1
⟶
Subq
x1
x2
⟶
Subq
x0
x2
Theorem
4ee26..
:
∀ x0 x1 x2 .
d3786..
x0
(
d3786..
x1
x2
)
=
d3786..
(
d3786..
x0
x1
)
x2
(proof)
Theorem
ea6d0..
:
∀ x0 x1 .
Subq
(
d3786..
x0
x1
)
(
d3786..
x1
x0
)
(proof)
Theorem
d7325..
:
∀ x0 x1 .
d3786..
x0
x1
=
d3786..
x1
x0
(proof)
Theorem
c4edc..
:
∀ x0 .
d3786..
4a7ef..
x0
=
4a7ef..
(proof)
Theorem
5fecc..
:
∀ x0 .
d3786..
x0
4a7ef..
=
4a7ef..
(proof)
Theorem
5cae2..
:
∀ x0 .
d3786..
x0
x0
=
x0
(proof)
Known
3eff2..
:
∀ x0 x1 x2 .
prim1
x2
(
d3786..
x0
x1
)
⟶
and
(
prim1
x2
x0
)
(
prim1
x2
x1
)
Theorem
6deec..
:
∀ x0 x1 x2 .
d3786..
x0
(
0ac37..
x1
x2
)
=
0ac37..
(
d3786..
x0
x1
)
(
d3786..
x0
x2
)
(proof)
Theorem
62dce..
:
∀ x0 x1 x2 .
0ac37..
x0
(
d3786..
x1
x2
)
=
d3786..
(
0ac37..
x0
x1
)
(
0ac37..
x0
x2
)
(proof)
Theorem
a7b4e..
:
∀ x0 x1 .
Subq
x0
x1
=
(
d3786..
x0
x1
=
x0
)
(proof)
Theorem
87311..
:
∀ x0 x1 x2 .
nIn
x2
x0
⟶
nIn
x2
(
d3786..
x0
x1
)
(proof)
Theorem
ddd82..
:
∀ x0 x1 x2 .
nIn
x2
x1
⟶
nIn
x2
(
d3786..
x0
x1
)
(proof)
Theorem
e30c6..
:
∀ x0 x1 x2 .
nIn
x2
(
d3786..
x0
x1
)
⟶
or
(
nIn
x2
x0
)
(
nIn
x2
x1
)
(proof)
Definition
1ad11..
:=
λ x0 x1 .
1216a..
x0
(
λ x2 .
nIn
x2
x1
)
Known
b2421..
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
prim1
x2
x0
⟶
x1
x2
⟶
prim1
x2
(
1216a..
x0
x1
)
Theorem
753c4..
:
∀ x0 x1 x2 .
prim1
x2
x0
⟶
nIn
x2
x1
⟶
prim1
x2
(
1ad11..
x0
x1
)
(proof)
Known
6982e..
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
prim1
x2
(
1216a..
x0
x1
)
⟶
and
(
prim1
x2
x0
)
(
x1
x2
)
Theorem
017d4..
:
∀ x0 x1 x2 .
prim1
x2
(
1ad11..
x0
x1
)
⟶
and
(
prim1
x2
x0
)
(
nIn
x2
x1
)
(proof)
Theorem
26c23..
:
∀ x0 x1 x2 .
prim1
x2
(
1ad11..
x0
x1
)
⟶
prim1
x2
x0
(proof)
Known
ac5c1..
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
prim1
x2
(
1216a..
x0
x1
)
⟶
x1
x2
Theorem
62370..
:
∀ x0 x1 x2 .
prim1
x2
(
1ad11..
x0
x1
)
⟶
nIn
x2
x1
(proof)
Theorem
dcf34..
:
∀ x0 x1 .
Subq
(
1ad11..
x0
x1
)
x0
(proof)
Theorem
7855c..
:
∀ x0 x1 x2 .
Subq
x2
x1
⟶
Subq
(
1ad11..
x0
x1
)
(
1ad11..
x0
x2
)
(proof)
Theorem
c937b..
:
∀ x0 x1 x2 .
nIn
x2
x0
⟶
nIn
x2
(
1ad11..
x0
x1
)
(proof)
Theorem
225c6..
:
∀ x0 x1 x2 .
prim1
x2
x1
⟶
nIn
x2
(
1ad11..
x0
x1
)
(proof)
Known
dneg
:
∀ x0 : ο .
not
(
not
x0
)
⟶
x0
Theorem
fd21e..
:
∀ x0 x1 x2 .
nIn
x2
(
1ad11..
x0
x1
)
⟶
or
(
nIn
x2
x0
)
(
prim1
x2
x1
)
(proof)
Theorem
2c578..
:
∀ x0 .
1ad11..
x0
x0
=
4a7ef..
(proof)
Theorem
87924..
:
∀ x0 x1 x2 .
1ad11..
x0
(
d3786..
x1
x2
)
=
0ac37..
(
1ad11..
x0
x1
)
(
1ad11..
x0
x2
)
(proof)
Theorem
4639d..
:
∀ x0 x1 x2 .
1ad11..
x0
(
0ac37..
x1
x2
)
=
1ad11..
(
1ad11..
x0
x1
)
x2
(proof)
Theorem
c89f2..
:
∀ x0 x1 x2 .
1ad11..
(
d3786..
x0
x1
)
x2
=
d3786..
x0
(
1ad11..
x1
x2
)
(proof)
Theorem
f1c06..
:
∀ x0 x1 x2 .
1ad11..
(
0ac37..
x0
x1
)
x2
=
0ac37..
(
1ad11..
x0
x2
)
(
1ad11..
x1
x2
)
(proof)
Theorem
60021..
:
∀ x0 x1 x2 .
1ad11..
x0
(
1ad11..
x1
x2
)
=
0ac37..
(
1ad11..
x0
x1
)
(
d3786..
x0
x2
)
(proof)
Theorem
9a446..
:
∀ x0 .
1ad11..
4a7ef..
x0
=
4a7ef..
(proof)
Theorem
25164..
:
∀ x0 .
1ad11..
x0
4a7ef..
=
x0
(proof)
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