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Proofgold Proof
pf
Let x0 of type
ι
→
ι
be given.
Assume H0:
∀ x1 .
1eb0a..
x1
⟶
and
(
SNo
(
x0
x1
)
)
(
∃ x2 .
and
(
SNo
x2
)
(
∃ x3 .
and
(
SNo
x3
)
(
∃ x4 .
and
(
SNo
x4
)
(
∃ x5 .
and
(
SNo
x5
)
(
∃ x6 .
and
(
SNo
x6
)
(
∃ x7 .
and
(
SNo
x7
)
(
∃ x8 .
and
(
SNo
x8
)
(
x1
=
bbc71..
(
x0
x1
)
x2
x3
x4
x5
x6
x7
x8
)
)
)
)
)
)
)
)
.
Let x1 of type
ι
be given.
Assume H1:
1eb0a..
x1
.
Apply H0 with
x1
,
and
(
SNo
(
d4639..
x0
x1
)
)
(
∃ x2 .
and
(
SNo
x2
)
(
∃ x3 .
and
(
SNo
x3
)
(
∃ x4 .
and
(
SNo
x4
)
(
∃ x5 .
and
(
SNo
x5
)
(
∃ x6 .
and
(
SNo
x6
)
(
∃ x7 .
and
(
SNo
x7
)
(
x1
=
bbc71..
(
x0
x1
)
(
d4639..
x0
x1
)
x2
x3
x4
x5
x6
x7
)
)
)
)
)
)
)
leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2:
SNo
(
x0
x1
)
.
Assume H3:
∃ x2 .
and
(
SNo
x2
)
(
∃ x3 .
and
(
SNo
x3
)
(
∃ x4 .
and
(
SNo
x4
)
(
∃ x5 .
and
(
SNo
x5
)
(
∃ x6 .
and
(
SNo
x6
)
(
∃ x7 .
and
(
SNo
x7
)
(
∃ x8 .
and
(
SNo
x8
)
(
x1
=
bbc71..
(
x0
x1
)
x2
x3
x4
x5
x6
x7
x8
)
)
)
)
)
)
)
.
Apply Eps_i_ex with
λ x2 .
and
(
SNo
x2
)
(
∃ x3 .
and
(
SNo
x3
)
(
∃ x4 .
and
(
SNo
x4
)
(
∃ x5 .
and
(
SNo
x5
)
(
∃ x6 .
and
(
SNo
x6
)
(
∃ x7 .
and
(
SNo
x7
)
(
∃ x8 .
and
(
SNo
x8
)
(
x1
=
bbc71..
(
x0
x1
)
x2
x3
x4
x5
x6
x7
x8
)
)
)
)
)
)
)
.
The subproof is completed by applying H3.
■