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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
x0
∈
SNoS_
(
ordsucc
omega
)
.
Assume H1:
SNoLt
(
minus_SNo
omega
)
x0
.
Assume H2:
SNo
x0
.
Assume H3:
not
(
∀ x1 .
x1
∈
omega
⟶
∃ x2 .
and
(
x2
∈
SNoS_
omega
)
(
and
(
SNoLt
x2
x0
)
(
SNoLt
x0
(
add_SNo
x2
(
eps_
x1
)
)
)
)
)
.
Assume H4:
nIn
x0
(
SNoS_
omega
)
.
Assume H5:
∀ x1 .
nat_p
x1
⟶
∃ x2 .
and
(
x2
∈
SNoS_
omega
)
(
and
(
SNoLt
x2
x0
)
(
SNoLt
x0
(
add_SNo
x2
(
eps_
x1
)
)
)
)
.
Apply H3.
Let x1 of type
ι
be given.
Assume H6:
x1
∈
omega
.
Apply H5 with
x1
.
Apply omega_nat_p with
x1
.
The subproof is completed by applying H6.
■