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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, RealsStruct_leq x0 (field4 x0) (If_i (RealsStruct_leq x0 (field4 x0) x1) x1 (Field_minus (Field_of_RealsStruct x0) x1)) 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 . RealsStruct_leq x0 (field4 x0) x3 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H2.
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 . RealsStruct_leq x0 (field4 x0) x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply RealsStruct_minus_zero with x0, λ x2 x3 . RealsStruct_leq x0 x2 (Field_minus (Field_of_RealsStruct x0) x1) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply RealsStruct_minus_leq with x0, x1, field4 x0 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply RealsStruct_zero_In with x0.
The subproof is completed by applying H0.
Apply RealsStruct_leq_linear with x0, x1, field4 x0, RealsStruct_leq x0 x1 (field4 x0) leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply RealsStruct_zero_In with x0.
The subproof is completed by applying H0.
Assume H3: RealsStruct_leq x0 x1 (field4 x0).
The subproof is completed by applying H3.
Assume H3: RealsStruct_leq x0 (field4 x0) x1.
Apply FalseE with RealsStruct_leq x0 x1 (field4 x0).
Apply H2.
The subproof is completed by applying H3.
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