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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
RealsStruct
x0
.
Let x1 of type
ι
be given.
Assume H1:
x1
∈
field0
x0
.
Apply RealsStruct_minus_eq with
x0
,
x1
,
λ x2 x3 .
field2b
x0
x3
x3
=
field2b
x0
x1
x1
leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply Field_of_RealsStruct_2f with
x0
,
λ x2 x3 :
ι →
ι → ι
.
x2
(
explicit_Field_minus
(
field0
x0
)
(
ap
(
Field_of_RealsStruct
x0
)
3
)
(
ap
(
Field_of_RealsStruct
x0
)
4
)
(
decode_b
(
ap
(
Field_of_RealsStruct
x0
)
1
)
)
(
decode_b
(
ap
(
Field_of_RealsStruct
x0
)
2
)
)
x1
)
(
explicit_Field_minus
(
field0
x0
)
(
ap
(
Field_of_RealsStruct
x0
)
3
)
(
ap
(
Field_of_RealsStruct
x0
)
4
)
(
decode_b
(
ap
(
Field_of_RealsStruct
x0
)
1
)
)
(
decode_b
(
ap
(
Field_of_RealsStruct
x0
)
2
)
)
x1
)
=
x2
x1
x1
leaving 2 subgoals.
The subproof is completed by applying H0.
Apply explicit_Field_minus_square with
field0
x0
,
ap
(
Field_of_RealsStruct
x0
)
3
,
ap
(
Field_of_RealsStruct
x0
)
4
,
decode_b
(
ap
(
Field_of_RealsStruct
x0
)
1
)
,
decode_b
(
ap
(
Field_of_RealsStruct
x0
)
2
)
,
x1
leaving 2 subgoals.
Apply explicit_Field_of_RealsStruct_2 with
x0
.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
■