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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
x0
∈
SNoS_
(
ordsucc
omega
)
.
Assume H1:
SNoL_omega
x0
=
0
⟶
∀ x1 : ο .
x1
.
Assume H2:
SNoR_omega
x0
=
0
⟶
∀ x1 : ο .
x1
.
Assume H3:
f8473..
(
SNoL_omega
x0
)
.
Assume H4:
f8473..
(
SNoR_omega
x0
)
.
Apply SepI with
SNoS_
(
ordsucc
omega
)
,
λ x1 .
and
(
and
(
and
(
SNoL_omega
x1
=
0
⟶
∀ x2 : ο .
x2
)
(
SNoR_omega
x1
=
0
⟶
∀ x2 : ο .
x2
)
)
(
f8473..
(
SNoL_omega
x1
)
)
)
(
f8473..
(
SNoR_omega
x1
)
)
,
x0
leaving 2 subgoals.
The subproof is completed by applying H0.
Apply and4I with
SNoL_omega
x0
=
0
⟶
∀ x1 : ο .
x1
,
SNoR_omega
x0
=
0
⟶
∀ x1 : ο .
x1
,
f8473..
(
SNoL_omega
x0
)
,
f8473..
(
SNoR_omega
x0
)
leaving 4 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
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