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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
RealsStruct
x0
.
Apply explicit_Reals_E with
field0
x0
,
field4
x0
,
RealsStruct_one
x0
,
field1b
x0
,
field2b
x0
,
RealsStruct_leq
x0
,
∀ x1 .
x1
∈
field0
x0
⟶
RealsStruct_leq
x0
x1
x1
leaving 2 subgoals.
Assume H1:
explicit_Reals
(
field0
x0
)
(
field4
x0
)
(
RealsStruct_one
x0
)
(
field1b
x0
)
(
field2b
x0
)
(
RealsStruct_leq
x0
)
.
Assume H2:
explicit_OrderedField
(
field0
x0
)
(
field4
x0
)
(
RealsStruct_one
x0
)
(
field1b
x0
)
(
field2b
x0
)
(
RealsStruct_leq
x0
)
.
Assume H3:
∀ x1 .
x1
∈
field0
x0
⟶
∀ x2 .
x2
∈
field0
x0
⟶
lt
(
field0
x0
)
(
field4
x0
)
(
RealsStruct_one
x0
)
(
field1b
x0
)
(
field2b
x0
)
(
RealsStruct_leq
x0
)
(
field4
x0
)
x1
⟶
RealsStruct_leq
x0
(
field4
x0
)
x2
⟶
∃ x3 .
and
(
x3
∈
Sep
(
field0
x0
)
(
natOfOrderedField_p
(
field0
x0
)
(
field4
x0
)
(
RealsStruct_one
x0
)
(
field1b
x0
)
(
field2b
x0
)
(
RealsStruct_leq
x0
)
)
)
(
RealsStruct_leq
x0
x2
(
field2b
x0
x3
x1
)
)
.
Assume H4:
∀ x1 .
x1
∈
setexp
(
field0
x0
)
(
Sep
(
field0
x0
)
(
natOfOrderedField_p
(
field0
x0
)
(
field4
x0
)
(
RealsStruct_one
x0
)
(
field1b
x0
)
(
field2b
x0
)
(
RealsStruct_leq
x0
)
)
)
⟶
∀ x2 .
x2
∈
setexp
(
field0
x0
)
(
Sep
(
field0
x0
)
(
natOfOrderedField_p
(
field0
x0
)
(
field4
x0
)
(
RealsStruct_one
x0
)
(
field1b
x0
)
(
field2b
x0
)
(
RealsStruct_leq
x0
)
)
)
⟶
(
∀ x3 .
x3
∈
Sep
(
field0
x0
)
(
natOfOrderedField_p
(
field0
x0
)
(
field4
x0
)
(
RealsStruct_one
x0
)
(
field1b
x0
)
(
field2b
x0
)
(
RealsStruct_leq
x0
)
)
⟶
and
(
and
(
RealsStruct_leq
x0
(
ap
x1
x3
)
(
ap
x2
x3
)
)
(
RealsStruct_leq
x0
(
ap
x1
x3
)
(
ap
x1
(
field1b
x0
x3
(
RealsStruct_one
x0
)
)
)
)
)
(
RealsStruct_leq
x0
(
ap
x2
(
field1b
x0
x3
(
RealsStruct_one
x0
)
)
)
(
ap
x2
x3
)
)
)
⟶
∃ x3 .
and
(
x3
∈
field0
x0
)
(
∀ x4 .
x4
∈
Sep
(
field0
x0
)
(
natOfOrderedField_p
(
field0
x0
)
(
field4
x0
)
(
RealsStruct_one
x0
)
(
field1b
x0
)
(
field2b
x0
)
(
RealsStruct_leq
x0
)
)
⟶
and
(
RealsStruct_leq
x0
(
ap
x1
x4
)
x3
)
(
RealsStruct_leq
x0
x3
(
ap
x2
x4
)
)
)
.
Apply explicit_OrderedField_leq_refl with
field0
x0
,
field4
x0
,
RealsStruct_one
x0
,
field1b
x0
,
field2b
x0
,
RealsStruct_leq
x0
.
The subproof is completed by applying H2.
Apply RealsStruct_explicit_Reals with
...
.
...
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