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Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Assume H0:
radical_field_extension
x0
x1
.
Let x2 of type
ο
be given.
Assume H1:
Field
x0
⟶
Field
x1
⟶
subfield
x0
x1
⟶
∀ x3 .
x3
∈
omega
⟶
∀ x4 .
ap
x4
0
=
x0
⟶
ap
x4
x3
=
x1
⟶
(
∀ x5 .
x5
∈
ordsucc
x3
⟶
Field
(
ap
x4
x5
)
)
⟶
(
∀ x5 .
x5
∈
ordsucc
x3
⟶
∀ x6 .
x6
∈
ordsucc
x5
⟶
subfield
(
ap
x4
x6
)
(
ap
x4
x5
)
)
⟶
(
∀ x5 .
x5
∈
x3
⟶
∃ x6 .
and
(
x6
∈
field0
(
ap
x4
(
ordsucc
x5
)
)
)
(
∃ x7 .
and
(
x7
∈
omega
)
(
and
(
CRing_with_id_omega_exp
(
ap
x4
(
ordsucc
x5
)
)
x6
x7
∈
field0
(
ap
x4
x5
)
)
(
Field_extension_by_1
(
ap
x4
x5
)
(
ap
x4
(
ordsucc
x5
)
)
x6
)
)
)
)
⟶
x2
.
Apply H0 with
x2
.
Let x3 of type
ι
be given.
Assume H2:
(
λ x4 .
and
(
x4
∈
omega
)
(
∃ x5 .
and
(
and
(
and
(
ap
x5
0
=
x0
)
(
ap
x5
x4
=
x1
)
)
(
∀ x6 .
x6
∈
ordsucc
x4
⟶
Field
(
ap
x5
x6
)
)
)
(
∀ x6 .
x6
∈
x4
⟶
∃ x7 .
and
(
x7
∈
field0
(
ap
x5
(
ordsucc
x6
)
)
)
(
∃ x8 .
and
(
x8
∈
omega
)
(
and
(
CRing_with_id_omega_exp
(
ap
x5
(
ordsucc
x6
)
)
x7
x8
∈
field0
(
ap
x5
x6
)
)
(
Field_extension_by_1
(
ap
x5
x6
)
(
ap
x5
(
ordsucc
x6
)
)
x7
)
)
)
)
)
)
x3
.
Apply H2 with
x2
.
Assume H3:
x3
∈
omega
.
Assume H4:
∃ x4 .
and
(
and
(
and
(
ap
x4
0
=
x0
)
(
ap
x4
x3
=
x1
)
)
(
∀ x5 .
x5
∈
ordsucc
x3
⟶
Field
(
ap
x4
x5
)
)
)
(
∀ x5 .
x5
∈
x3
⟶
∃ x6 .
and
(
x6
∈
field0
(
ap
x4
(
ordsucc
x5
)
)
)
(
∃ x7 .
and
(
x7
∈
omega
)
(
and
(
CRing_with_id_omega_exp
(
ap
x4
(
ordsucc
x5
)
)
x6
x7
∈
field0
(
ap
x4
x5
)
)
(
Field_extension_by_1
(
ap
x4
x5
)
(
ap
x4
(
ordsucc
x5
)
)
x6
)
)
)
)
.
Apply H4 with
x2
.
Let x4 of type
ι
be given.
Assume H5:
(
λ x5 .
and
(
and
(
and
(
ap
x5
0
=
x0
)
(
ap
x5
x3
=
x1
)
)
(
∀ x6 .
x6
∈
ordsucc
x3
⟶
Field
(
ap
x5
x6
)
)
)
(
∀ x6 .
...
⟶
∃ x7 .
and
(
x7
∈
field0
(
ap
x5
...
)
)
...
)
)
...
.
...
■