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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.
Let x2 of type ι be given.
Assume H2: x2 ∈ field0 x0.
Apply RealsStruct_lt_trich_impred with x0, x1, x2, or (RealsStruct_leq x0 x1 x2) (RealsStruct_lt x0 x2 x1) leaving 6 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Assume H3: RealsStruct_lt x0 x1 x2.
Apply H3 with or (RealsStruct_leq x0 x1 x2) (RealsStruct_lt x0 x2 x1).
Assume H4: RealsStruct_leq x0 x1 x2.
Assume H5: x1 = x2 ⟶ ∀ x3 : ο . x3.
Apply orIL with RealsStruct_leq x0 x1 x2, RealsStruct_lt x0 x2 x1.
The subproof is completed by applying H4.
Assume H3: x1 = x2.
Apply orIL with RealsStruct_leq x0 x1 x2, RealsStruct_lt x0 x2 x1.
Apply H3 with λ x3 x4 . RealsStruct_leq x0 x1 x3.
Apply RealsStruct_leq_refl with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying orIR with RealsStruct_leq x0 x1 x2, RealsStruct_lt x0 x2 x1.
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