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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
x0
⊆
SNoS_
omega
.
Assume H1:
∀ x1 .
x1
∈
x0
⟶
∃ x2 .
and
(
x2
∈
omega
)
(
∀ x3 .
x3
∈
SNoS_
omega
⟶
SNoLt
(
add_SNo
x1
(
minus_SNo
(
eps_
x2
)
)
)
x3
⟶
SNoLt
x3
(
add_SNo
x1
(
eps_
x2
)
)
⟶
x3
∈
x0
)
.
Apply andI with
x0
⊆
SNoS_
omega
,
∀ x1 .
x1
∈
x0
⟶
∃ x2 .
and
(
x2
∈
omega
)
(
∀ x3 .
x3
∈
SNoS_
omega
⟶
SNoLt
(
add_SNo
x1
(
minus_SNo
(
eps_
x2
)
)
)
x3
⟶
SNoLt
x3
(
add_SNo
x1
(
eps_
x2
)
)
⟶
x3
∈
x0
)
leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
■