Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ι be given.
Apply H4 with
RealsStruct_lt x0 x1 x3.
Assume H7: x1 = x2 ⟶ ∀ x4 : ο . x4.
Apply andI with
RealsStruct_leq x0 x1 x3,
x1 = x3 ⟶ ∀ x4 : ο . x4 leaving 2 subgoals.
Apply RealsStruct_leq_tra with
x0,
x1,
x2,
x3 leaving 6 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 H3.
The subproof is completed by applying H6.
The subproof is completed by applying H5.
Assume H8: x1 = x3.
Apply RealsStruct_lt_leq_asym with
x0,
x1,
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 H4.
Apply H8 with
λ x4 x5 . RealsStruct_leq x0 x2 x5.
The subproof is completed by applying H5.