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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
RealsStruct
x0
.
Apply RealsStruct_eta with
x0
,
λ x1 x2 .
OrderedFieldStruct
x2
leaving 2 subgoals.
The subproof is completed by applying H0.
Apply andI with
struct_b_b_r_e_e
(
pack_b_b_r_e_e
(
field0
x0
)
(
field1b
x0
)
(
field2b
x0
)
(
RealsStruct_leq
x0
)
(
field4
x0
)
(
RealsStruct_one
x0
)
)
,
unpack_b_b_r_e_e_o
(
pack_b_b_r_e_e
(
field0
x0
)
(
field1b
x0
)
(
field2b
x0
)
(
RealsStruct_leq
x0
)
(
field4
x0
)
(
RealsStruct_one
x0
)
)
(
λ x1 .
λ x2 x3 :
ι →
ι → ι
.
λ x4 :
ι →
ι → ο
.
λ x5 x6 .
explicit_OrderedField
x1
x5
x6
x2
x3
x4
)
leaving 2 subgoals.
Apply pack_struct_b_b_r_e_e_I with
field0
x0
,
field1b
x0
,
field2b
x0
,
RealsStruct_leq
x0
,
field4
x0
,
RealsStruct_one
x0
leaving 4 subgoals.
Apply RealsStruct_plus_clos with
x0
.
The subproof is completed by applying H0.
Apply RealsStruct_mult_clos with
x0
.
The subproof is completed by applying H0.
Apply RealsStruct_zero_In with
x0
.
The subproof is completed by applying H0.
Apply RealsStruct_one_In with
x0
.
The subproof is completed by applying H0.
Apply OrderedFieldStruct_unpack_eq with
field0
x0
,
field1b
x0
,
field2b
x0
,
RealsStruct_leq
x0
,
field4
x0
,
RealsStruct_one
x0
,
λ x1 x2 : ο .
x2
.
Apply explicit_OrderedField_of_RealsStruct with
x0
.
The subproof is completed by applying H0.
■