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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
Field
x0
.
Let x1 of type
ι
be given.
Let x2 of type
ι
be given.
Assume H1:
nIn
x1
(
K_field_0
x0
x0
)
.
Claim L2:
not
(
and
(
x1
∈
K_field_0
x0
x0
)
(
x2
∈
setminus
(
K_field_0
x0
x0
)
(
Sing
(
K_field_3
x0
x0
)
)
)
)
Assume H2:
and
(
x1
∈
K_field_0
x0
x0
)
(
x2
∈
setminus
(
K_field_0
x0
x0
)
(
Sing
(
K_field_3
x0
x0
)
)
)
.
Apply H2 with
False
.
Assume H3:
x1
∈
K_field_0
x0
x0
.
Assume H4:
x2
∈
setminus
(
K_field_0
x0
x0
)
(
Sing
(
K_field_3
x0
x0
)
)
.
Apply H1.
The subproof is completed by applying H3.
Apply If_i_0 with
and
(
x1
∈
K_field_0
x0
x0
)
(
x2
∈
setminus
(
K_field_0
x0
x0
)
(
Sing
(
K_field_3
x0
x0
)
)
)
,
prim0
(
λ x3 .
and
(
x3
∈
K_field_0
x0
x0
)
(
x1
=
K_field_2_b
x0
x0
x2
x3
)
)
,
0
.
The subproof is completed by applying L2.
■