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Definition
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Param
explicit_Field_minus
:
ι
→
ι
→
ι
→
(
ι
→
ι
→
ι
) →
(
ι
→
ι
→
ι
) →
ι
→
ι
Param
3b429..
:
ι
→
(
ι
→
ι
) →
(
ι
→
ι
→
ο
) →
CT2
ι
Param
True
:
ο
Known
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Definition
False
:=
∀ x0 : ο .
x0
Theorem
e6e02..
:
∀ x0 x1 x2 .
∀ x3 x4 :
ι →
ι → ι
.
∀ x5 :
ι →
ι → ο
.
∀ x6 :
ι →
ι → ι
.
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
prim1
(
x3
x7
x8
)
x0
)
⟶
prim1
x1
x0
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
prim1
(
x4
x7
x8
)
x0
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
∀ x9 .
prim1
x9
x0
⟶
x4
x7
(
x4
x8
x9
)
=
x4
(
x4
x7
x8
)
x9
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
x4
x7
x8
=
x4
x8
x7
)
⟶
prim1
x2
x0
⟶
(
∀ x7 .
prim1
x7
x0
⟶
(
x7
=
x1
⟶
∀ x8 : ο .
x8
)
⟶
∃ x8 .
and
(
prim1
x8
x0
)
(
x4
x7
x8
=
x2
)
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
∀ x9 .
prim1
x9
x0
⟶
x4
x7
(
x3
x8
x9
)
=
x3
(
x4
x7
x8
)
(
x4
x7
x9
)
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
prim1
(
explicit_Field_minus
x0
x1
x2
x3
x4
x7
)
x0
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
∀ x9 .
prim1
x9
x0
⟶
x4
(
x3
x7
x8
)
x9
=
x3
(
x4
x7
x9
)
(
x4
x8
x9
)
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
explicit_Field_minus
x0
x1
x2
x3
x4
(
x3
x7
x8
)
=
x3
(
explicit_Field_minus
x0
x1
x2
x3
x4
x7
)
(
explicit_Field_minus
x0
x1
x2
x3
x4
x8
)
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
x4
(
explicit_Field_minus
x0
x1
x2
x3
x4
x7
)
x8
=
explicit_Field_minus
x0
x1
x2
x3
x4
(
x4
x7
x8
)
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
x4
x7
(
explicit_Field_minus
x0
x1
x2
x3
x4
x8
)
=
explicit_Field_minus
x0
x1
x2
x3
x4
(
x4
x7
x8
)
)
⟶
(
∀ x7 .
prim1
x7
(
3b429..
x0
(
λ x8 .
x0
)
(
λ x8 x9 .
True
)
x6
)
⟶
prim1
(
prim0
(
λ x8 .
and
(
prim1
x8
x0
)
(
∃ x9 .
and
(
prim1
x9
x0
)
(
x7
=
x6
x8
x9
)
)
)
)
x0
)
⟶
(
∀ x7 .
prim1
x7
(
3b429..
x0
(
λ x8 .
x0
)
(
λ x8 x9 .
True
)
x6
)
⟶
prim1
(
prim0
(
λ x8 .
and
(
prim1
x8
x0
)
(
x7
=
x6
(
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
∃ x11 .
and
(
prim1
x11
x0
)
(
x7
=
x6
x10
x11
)
)
)
)
x8
)
)
)
x0
)
⟶
(
∀ x7 .
prim1
x7
(
3b429..
x0
(
λ x8 .
x0
)
(
λ x8 x9 .
True
)
x6
)
⟶
∀ x8 .
prim1
x8
(
3b429..
x0
(
λ x9 .
x0
)
(
λ x9 x10 .
True
)
x6
)
⟶
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
∃ x11 .
and
(
prim1
x11
x0
)
(
x7
=
x6
x10
x11
)
)
)
=
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
∃ x11 .
and
(
prim1
x11
x0
)
(
x8
=
x6
x10
x11
)
)
)
⟶
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
x7
=
x6
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x7
=
x6
x12
x13
)
)
)
)
x10
)
)
=
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
x8
=
x6
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x8
=
x6
x12
x13
)
)
)
)
x10
)
)
⟶
x7
=
x8
)
⟶
(
∀ x7 .
prim1
x7
(
3b429..
x0
(
λ x8 .
x0
)
(
λ x8 x9 .
True
)
x6
)
⟶
∀ x8 .
prim1
x8
(
3b429..
x0
(
λ x9 .
x0
)
(
λ x9 x10 .
True
)
x6
)
⟶
prim1
(
x6
(
x3
(
x4
(
prim0
(
λ x9 .
and
(
prim1
x9
x0
)
(
∃ x10 .
and
(
prim1
x10
x0
)
(
x7
=
x6
x9
x10
)
)
)
)
(
prim0
(
λ x9 .
and
(
prim1
x9
x0
)
(
∃ x10 .
and
(
prim1
x10
x0
)
(
x8
=
x6
x9
x10
)
)
)
)
)
(
explicit_Field_minus
x0
x1
x2
x3
x4
(
x4
(
prim0
(
λ x9 .
and
(
prim1
x9
x0
)
(
x7
=
x6
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
∃ x12 .
and
(
prim1
x12
x0
)
(
x7
=
x6
x11
x12
)
)
)
)
x9
)
)
)
(
prim0
(
λ x9 .
and
(
prim1
x9
x0
)
(
x8
=
x6
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
∃ x12 .
and
(
prim1
x12
x0
)
(
x8
=
x6
x11
x12
)
)
)
)
x9
)
)
)
)
)
)
(
x3
(
x4
(
prim0
(
λ x9 .
and
(
prim1
x9
x0
)
(
∃ x10 .
and
(
prim1
x10
x0
)
(
x7
=
x6
x9
x10
)
)
)
)
(
prim0
(
λ x9 .
and
(
prim1
x9
x0
)
(
x8
=
x6
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
∃ x12 .
and
(
prim1
x12
x0
)
(
x8
=
x6
x11
x12
)
)
)
)
x9
)
)
)
)
(
x4
(
prim0
(
λ x9 .
and
(
prim1
x9
x0
)
(
x7
=
x6
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
∃ x12 .
and
(
prim1
x12
x0
)
(
x7
=
x6
x11
x12
)
)
)
)
x9
)
)
)
(
prim0
(
λ x9 .
and
(
prim1
x9
x0
)
(
∃ x10 .
and
(
prim1
x10
x0
)
(
x8
=
x6
x9
x10
)
)
)
)
)
)
)
(
3b429..
x0
(
λ x9 .
x0
)
(
λ x9 x10 .
True
)
x6
)
)
⟶
(
∀ x7 .
prim1
x7
(
3b429..
x0
(
λ x8 .
x0
)
(
λ x8 x9 .
True
)
x6
)
⟶
∀ x8 .
prim1
x8
(
3b429..
x0
(
λ x9 .
x0
)
(
λ x9 x10 .
True
)
x6
)
⟶
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
∃ x11 .
and
(
prim1
x11
x0
)
(
x6
(
x3
(
x4
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
∃ x15 .
and
(
prim1
x15
x0
)
(
x7
=
x6
x14
x15
)
)
)
)
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
∃ x15 .
and
(
prim1
x15
x0
)
(
x8
=
x6
x14
x15
)
)
)
)
)
(
explicit_Field_minus
x0
x1
x2
x3
x4
(
x4
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
x7
=
x6
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
∃ x17 .
and
(
prim1
x17
x0
)
(
x7
=
x6
x16
x17
)
)
)
)
x14
)
)
)
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
x8
=
x6
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
∃ x17 .
and
(
prim1
x17
x0
)
(
x8
=
x6
x16
x17
)
)
)
)
x14
)
)
)
)
)
)
(
x3
(
x4
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
∃ x15 .
and
(
prim1
x15
x0
)
(
x7
=
x6
x14
x15
)
)
)
)
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
x8
=
x6
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
∃ x17 .
and
(
prim1
x17
x0
)
(
x8
=
x6
x16
x17
)
)
)
)
x14
)
)
)
)
(
x4
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
x7
=
x6
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
∃ x17 .
and
(
prim1
x17
x0
)
(
x7
=
x6
x16
x17
)
)
)
)
x14
)
)
)
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
∃ x15 .
and
(
prim1
x15
x0
)
(
x8
=
x6
x14
x15
)
)
)
)
)
)
=
x6
x10
x11
)
)
)
=
x3
(
x4
(
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
∃ x11 .
and
(
prim1
x11
x0
)
(
x7
=
x6
x10
x11
)
)
)
)
(
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
∃ x11 .
and
(
prim1
x11
x0
)
(
x8
=
x6
x10
x11
)
)
)
)
)
(
explicit_Field_minus
x0
x1
x2
x3
x4
(
x4
(
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
x7
=
x6
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x7
=
x6
x12
x13
)
)
)
)
x10
)
)
)
(
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
x8
=
x6
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x8
=
x6
x12
x13
)
)
)
)
x10
)
)
)
)
)
)
⟶
(
∀ x7 .
prim1
x7
(
3b429..
x0
(
λ x8 .
x0
)
(
λ x8 x9 .
True
)
x6
)
⟶
∀ x8 .
prim1
x8
(
3b429..
x0
(
λ x9 .
x0
)
(
λ x9 x10 .
True
)
x6
)
⟶
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
x6
(
x3
(
x4
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x7
=
x6
x12
x13
)
)
)
)
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x8
=
x6
x12
x13
)
)
)
)
)
(
explicit_Field_minus
x0
x1
x2
x3
x4
(
x4
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
x7
=
x6
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
∃ x15 .
and
(
prim1
x15
x0
)
(
x7
=
x6
x14
x15
)
)
)
)
x12
)
)
)
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
x8
=
x6
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
∃ x15 .
and
(
prim1
x15
x0
)
(
x8
=
x6
x14
x15
)
)
)
)
x12
)
)
)
)
)
)
(
x3
(
x4
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x7
=
x6
x12
x13
)
)
)
)
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
x8
=
x6
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
∃ x15 .
and
(
prim1
x15
x0
)
(
x8
=
x6
x14
x15
)
)
)
)
x12
)
)
)
)
(
x4
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
x7
=
x6
(
prim0
(
λ x14 .
and
(
prim1
x14
x0
)
(
∃ x15 .
and
(
prim1
x15
x0
)
(
x7
=
x6
x14
x15
)
)
)
)
x12
)
)
)
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x8
=
x6
x12
x13
)
)
)
)
)
)
=
x6
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x6
(
x3
(
x4
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
∃ x17 .
and
(
prim1
x17
x0
)
(
x7
=
x6
x16
x17
)
)
)
)
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
∃ x17 .
and
(
prim1
x17
x0
)
(
x8
=
x6
x16
x17
)
)
)
)
)
(
explicit_Field_minus
x0
x1
x2
x3
x4
(
x4
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
x7
=
x6
(
prim0
(
λ x18 .
and
(
prim1
x18
x0
)
(
∃ x19 .
and
(
prim1
x19
x0
)
(
x7
=
x6
x18
x19
)
)
)
)
x16
)
)
)
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
x8
=
x6
(
prim0
(
λ x18 .
and
(
prim1
x18
x0
)
(
∃ x19 .
and
(
prim1
x19
x0
)
(
x8
=
x6
x18
x19
)
)
)
)
x16
)
)
)
)
)
)
(
x3
(
x4
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
∃ x17 .
and
(
prim1
x17
x0
)
(
x7
=
x6
x16
x17
)
)
)
)
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
x8
=
x6
(
prim0
(
λ x18 .
and
(
prim1
x18
x0
)
(
∃ x19 .
and
(
prim1
x19
x0
)
(
x8
=
x6
x18
x19
)
)
)
)
x16
)
)
)
)
(
x4
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
x7
=
x6
(
prim0
(
λ x18 .
and
(
prim1
x18
x0
)
(
∃ x19 .
and
(
prim1
x19
x0
)
(
x7
=
x6
x18
x19
)
)
)
)
x16
)
)
)
(
prim0
(
λ x16 .
and
(
prim1
x16
x0
)
(
∃ x17 .
and
(
prim1
x17
x0
)
(
x8
=
x6
x16
x17
)
)
)
)
)
)
=
x6
x12
x13
)
)
)
)
x10
)
)
=
x3
(
x4
(
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
∃ x11 .
and
(
prim1
x11
x0
)
(
x7
=
x6
x10
x11
)
)
)
)
(
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
x8
=
x6
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x8
=
x6
x12
x13
)
)
)
)
x10
)
)
)
)
(
x4
(
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
x7
=
x6
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x7
=
x6
x12
x13
)
)
)
)
x10
)
)
)
(
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
∃ x11 .
and
(
prim1
x11
x0
)
(
x8
=
x6
x10
x11
)
)
)
)
)
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
prim1
(
x6
x7
x8
)
(
3b429..
x0
(
λ x9 .
x0
)
(
λ x9 x10 .
True
)
x6
)
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
∃ x11 .
and
(
prim1
x11
x0
)
(
x6
x7
x8
=
x6
x10
x11
)
)
)
=
x7
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
prim0
(
λ x10 .
and
(
prim1
x10
x0
)
(
x6
x7
x8
=
x6
(
prim0
(
λ x12 .
and
(
prim1
x12
x0
)
(
∃ x13 .
and
(
prim1
x13
x0
)
(
x6
x7
x8
=
x6
x12
x13
)
)
)
)
x10
)
)
=
x8
)
⟶
prim1
(
x6
x1
x1
)
(
3b429..
x0
(
λ x7 .
x0
)
(
λ x7 x8 .
True
)
x6
)
⟶
prim1
(
x6
x2
x1
)
(
3b429..
x0
(
λ x7 .
x0
)
(
λ x7 x8 .
True
)
x6
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
explicit_Field_minus
x0
x1
x2
x3
x4
(
explicit_Field_minus
x0
x1
x2
x3
x4
x7
)
=
x7
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
x3
(
explicit_Field_minus
x0
x1
x2
x3
x4
x7
)
x7
=
x1
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
x4
x1
x7
=
x1
)
⟶
(
∀ x7 .
prim1
x7
x0
⟶
∀ x8 .
prim1
x8
x0
⟶
x3
(
x4
x7
x7
)
(
x4
x8
x8
)
=
x1
⟶
and
(
x7
=
x1
)
(
x8
=
x1
)
)
⟶
∀ x7 .
prim1
x7
(
3b429..
x0
(
λ x8 .
x0
)
(
λ x8 x9 .
True
)
x6
)
⟶
(
x7
=
x6
x1
x1
⟶
∀ x8 : ο .
x8
)
⟶
∃ x8 .
and
(
prim1
x8
(
3b429..
x0
(
λ x10 .
x0
)
(
λ x10 x11 .
True
)
x6
)
)
(
x6
(
x3
(
x4
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
∃ x12 .
and
(
prim1
x12
x0
)
(
x7
=
x6
x11
x12
)
)
)
)
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
∃ x12 .
and
(
prim1
x12
x0
)
(
x8
=
x6
x11
x12
)
)
)
)
)
(
explicit_Field_minus
x0
x1
x2
x3
x4
(
x4
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
x7
=
x6
(
prim0
(
λ x13 .
and
(
prim1
x13
x0
)
(
∃ x14 .
and
(
prim1
x14
x0
)
(
x7
=
x6
x13
x14
)
)
)
)
x11
)
)
)
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
x8
=
x6
(
prim0
(
λ x13 .
and
(
prim1
x13
x0
)
(
∃ x14 .
and
(
prim1
x14
x0
)
(
x8
=
x6
x13
x14
)
)
)
)
x11
)
)
)
)
)
)
(
x3
(
x4
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
∃ x12 .
and
(
prim1
x12
x0
)
(
x7
=
x6
x11
x12
)
)
)
)
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
x8
=
x6
(
prim0
(
λ x13 .
and
(
prim1
x13
x0
)
(
∃ x14 .
and
(
prim1
x14
x0
)
(
x8
=
x6
x13
x14
)
)
)
)
x11
)
)
)
)
(
x4
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
x7
=
x6
(
prim0
(
λ x13 .
and
(
prim1
x13
x0
)
(
∃ x14 .
and
(
prim1
x14
x0
)
(
x7
=
x6
x13
x14
)
)
)
)
x11
)
)
)
(
prim0
(
λ x11 .
and
(
prim1
x11
x0
)
(
∃ x12 .
and
(
prim1
x12
x0
)
(
x8
=
x6
x11
x12
)
)
)
)
)
)
=
x6
x2
x1
)
...