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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 Field_of_RealsStruct_0 with
x0
,
λ x2 x3 .
If_i
(
x1
∈
x3
)
(
explicit_Field_minus
x3
(
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
)
0
=
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
.
Apply If_i_1 with
x1
∈
field0
x0
,
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
,
0
.
The subproof is completed by applying H1.
■