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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
x0
∈
real
.
Apply real_E with
x0
,
SNo
x0
leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1:
SNo
x0
.
Assume H2:
SNoLev
x0
∈
ordsucc
omega
.
Assume H3:
x0
∈
SNoS_
(
ordsucc
omega
)
.
Assume H4:
SNoLt
(
minus_SNo
omega
)
x0
.
Assume H5:
SNoLt
x0
omega
.
Assume H6:
∀ x1 .
x1
∈
SNoS_
omega
⟶
(
∀ x2 .
x2
∈
omega
⟶
SNoLt
(
abs_SNo
(
add_SNo
x1
(
minus_SNo
x0
)
)
)
(
eps_
x2
)
)
⟶
x1
=
x0
.
Assume H7:
∀ x1 .
x1
∈
omega
⟶
∃ x2 .
and
(
x2
∈
SNoS_
omega
)
(
and
(
SNoLt
x2
x0
)
(
SNoLt
x0
(
add_SNo
x2
(
eps_
x1
)
)
)
)
.
The subproof is completed by applying H1.
■