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.
Let x4 of type ι → ι → ο be given.
Assume H4: ∀ x5 x6 . x4 x5 x6 ⟶ x4 x6 x5.
Apply unknownprop_b2fe1166cc90aada9c2da7a2f96ec027f60d52a2799891b4e0a891d3402a8b5f with
x2,
x3,
{x5 ∈ add_nat x2 x3|not (x4 x5 (add_nat x2 x3))},
{x5 ∈ add_nat x2 x3|x4 x5 (add_nat x2 x3)},
or (∃ x5 . and (x5 ⊆ ordsucc (add_nat x2 x3)) (and (atleastp (ordsucc x0) x5) (∀ x6 . x6 ∈ x5 ⟶ ∀ x7 . x7 ∈ x5 ⟶ (x6 = x7 ⟶ ∀ x8 : ο . x8) ⟶ x4 x6 x7))) (∃ x5 . and (x5 ⊆ ordsucc (add_nat x2 x3)) (and (atleastp (ordsucc x1) x5) (∀ x6 . x6 ∈ x5 ⟶ ∀ x7 . x7 ∈ x5 ⟶ (x6 = x7 ⟶ ∀ x8 : ο . x8) ⟶ not (x4 x6 x7)))) leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying L6.