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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
x0
∈
SNoS_
(
ordsucc
omega
)
.
Assume H1:
x0
=
omega
⟶
∀ x1 : ο .
x1
.
Assume H2:
x0
=
minus_SNo
omega
⟶
∀ x1 : ο .
x1
.
Assume H3:
∀ x1 .
x1
∈
SNoS_
omega
⟶
(
∀ x2 .
x2
∈
omega
⟶
SNoLt
(
abs_SNo
(
add_SNo
x1
(
minus_SNo
x0
)
)
)
(
eps_
x2
)
)
⟶
x1
=
x0
.
Apply SepI with
SNoS_
(
ordsucc
omega
)
,
λ x1 .
and
(
and
(
x1
=
omega
⟶
∀ x2 : ο .
x2
)
(
x1
=
minus_SNo
omega
⟶
∀ x2 : ο .
x2
)
)
(
∀ x2 .
x2
∈
SNoS_
omega
⟶
(
∀ x3 .
x3
∈
omega
⟶
SNoLt
(
abs_SNo
(
add_SNo
x2
(
minus_SNo
x1
)
)
)
(
eps_
x3
)
)
⟶
x2
=
x1
)
,
x0
leaving 2 subgoals.
The subproof is completed by applying H0.
Apply and3I with
x0
=
omega
⟶
∀ x1 : ο .
x1
,
x0
=
minus_SNo
omega
⟶
∀ x1 : ο .
x1
,
∀ x1 .
x1
∈
SNoS_
omega
⟶
(
∀ x2 .
x2
∈
omega
⟶
SNoLt
(
abs_SNo
(
add_SNo
x1
(
minus_SNo
x0
)
)
)
(
eps_
x2
)
)
⟶
x1
=
x0
leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
■