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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
x0
∈
87ab7..
.
Let x1 of type
ο
be given.
Assume H1:
x0
∈
SNoS_
(
ordsucc
omega
)
⟶
(
SNoL_omega
x0
=
0
⟶
∀ x2 : ο .
x2
)
⟶
(
SNoR_omega
x0
=
0
⟶
∀ x2 : ο .
x2
)
⟶
f8473..
(
SNoL_omega
x0
)
⟶
f8473..
(
SNoR_omega
x0
)
⟶
x1
.
Apply SepE with
SNoS_
(
ordsucc
omega
)
,
λ x2 .
and
(
and
(
and
(
SNoL_omega
x2
=
0
⟶
∀ x3 : ο .
x3
)
(
SNoR_omega
x2
=
0
⟶
∀ x3 : ο .
x3
)
)
(
f8473..
(
SNoL_omega
x2
)
)
)
(
f8473..
(
SNoR_omega
x2
)
)
,
x0
,
x1
leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H2:
x0
∈
SNoS_
(
ordsucc
omega
)
.
Assume H3:
and
(
and
(
and
(
SNoL_omega
x0
=
0
⟶
∀ x2 : ο .
x2
)
(
SNoR_omega
x0
=
0
⟶
∀ x2 : ο .
x2
)
)
(
f8473..
(
SNoL_omega
x0
)
)
)
(
f8473..
(
SNoR_omega
x0
)
)
.
Apply and4E with
SNoL_omega
x0
=
0
⟶
∀ x2 : ο .
x2
,
SNoR_omega
x0
=
0
⟶
∀ x2 : ο .
x2
,
f8473..
(
SNoL_omega
x0
)
,
f8473..
(
SNoR_omega
x0
)
,
x1
leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H1.
The subproof is completed by applying H2.
■