Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply RealsStruct_minus_clos with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
Apply RealsStruct_plus_clos with
x0,
x1,
Field_minus (Field_of_RealsStruct x0) x2 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying L4.
Apply RealsStruct_abs_zero_inv with
x0,
field1b x0 x1 (Field_minus (Field_of_RealsStruct x0) x2) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L5.
The subproof is completed by applying H3.
Apply RealsStruct_plus_cancelR with
x0,
x1,
x2,
Field_minus (Field_of_RealsStruct x0) x2 leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying L4.
Claim L7: ∀ x5 : ι → ο . x5 y4 ⟶ x5 y3
Let x5 of type ι → ο be given.
Apply L6 with
λ x6 . x5.
set y6 to be λ x6 . x5
Apply RealsStruct_minus_R with
x2,
y4,
λ x7 x8 . y6 x8 x7 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
The subproof is completed by applying H7.
Let x5 of type ι → ι → ο be given.
Apply L7 with
λ x6 . x5 x6 y4 ⟶ x5 y4 x6.
Assume H8: x5 y4 y4.
The subproof is completed by applying H8.