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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 xm with
RealsStruct_leq
x0
(
field4
x0
)
x1
,
If_i
(
RealsStruct_leq
x0
(
field4
x0
)
x1
)
x1
(
Field_minus
(
Field_of_RealsStruct
x0
)
x1
)
∈
field0
x0
leaving 2 subgoals.
Assume H2:
RealsStruct_leq
x0
(
field4
x0
)
x1
.
Apply If_i_1 with
RealsStruct_leq
x0
(
field4
x0
)
x1
,
x1
,
Field_minus
(
Field_of_RealsStruct
x0
)
x1
,
λ x2 x3 .
x3
∈
field0
x0
leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.
Assume H2:
not
(
RealsStruct_leq
x0
(
field4
x0
)
x1
)
.
Apply If_i_0 with
RealsStruct_leq
x0
(
field4
x0
)
x1
,
x1
,
Field_minus
(
Field_of_RealsStruct
x0
)
x1
,
λ x2 x3 .
x3
∈
field0
x0
leaving 2 subgoals.
The subproof is completed by applying H2.
Apply RealsStruct_minus_clos with
x0
,
x1
leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
■