Let x0 of type ι be given.
Let x1 of type ι be given.
Apply RealsStruct_minus_clos with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply RealsStruct_mult_clos with
x0,
Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0),
x1 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply RealsStruct_minus_one_In with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply RealsStruct_plus_cancelL with
x0,
x1,
Field_minus (Field_of_RealsStruct x0) x1,
field2b x0 (Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0)) x1 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying L2.
The subproof is completed by applying L3.
Apply RealsStruct_minus_R with
x0,
x1,
λ x2 x3 . x3 = field1b x0 x1 (field2b x0 (Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0)) x1) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Let x2 of type ι → ι → ο be given.
Apply RealsStruct_mult_com with
x0,
Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0),
x1,
λ x3 x4 . field1b x0 x1 x4 = field4 x0,
λ x3 x4 . x2 x4 x3 leaving 4 subgoals.
The subproof is completed by applying H0.
Apply RealsStruct_minus_one_In with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply RealsStruct_one_L with
x0,
x1,
λ x3 x4 . field1b x0 x3 (field2b x0 x1 (Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0))) = field4 x0 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply RealsStruct_mult_com with
x0,
RealsStruct_one x0,
x1,
λ x3 x4 . field1b x0 x4 (field2b x0 x1 (Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0))) = field4 x0 leaving 4 subgoals.
The subproof is completed by applying H0.
Apply RealsStruct_one_In with
x0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply RealsStruct_distr_L with
x0,
x1,
RealsStruct_one x0,
Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0),
λ x3 x4 . x3 = field4 x0 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply RealsStruct_one_In with
x0.
The subproof is completed by applying H0.
Apply RealsStruct_minus_one_In with
x0.
The subproof is completed by applying H0.
Apply RealsStruct_minus_R with
x0,
RealsStruct_one x0,
λ x3 x4 . field2b x0 x1 x4 = field4 x0 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply RealsStruct_one_In with
x0.
The subproof is completed by applying H0.
Apply RealsStruct_zero_multR with
x0,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.