Let x0 of type ι be given.
Let x1 of type ι be given.
set y3 to be x1
Claim L2: ∀ x4 : ι → ο . x4 y3 ⟶ x4 y2
Let x4 of type ι → ο be given.
Assume H2: x4 y2.
set y5 to be λ x5 . x4
Apply add_SNo_assoc with
y2,
y3,
minus_SNo y3,
λ x6 x7 . y5 x7 x6 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNo_minus_SNo with
y3.
The subproof is completed by applying H1.
Claim L3: ∀ x8 : ι → ο . x8 y7 ⟶ x8 y6
Let x8 of type ι → ο be given.
set y9 to be λ x9 . x8
Apply add_SNo_minus_SNo_rinv with
y6,
λ x10 x11 . y9 (add_SNo y5 x10) (add_SNo y5 x11) leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
set y8 to be λ x8 . y7
Apply L3 with
λ x9 . y8 x9 y7 ⟶ y8 y7 x9 leaving 2 subgoals.
Assume H4: y8 y7 y7.
The subproof is completed by applying H4.
Apply add_SNo_0R with
y6,
λ x9 . y8 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying L3.
Let x4 of type ι → ι → ο be given.
Apply L2 with
λ x5 . x4 x5 y3 ⟶ x4 y3 x5.
Assume H3: x4 y3 y3.
The subproof is completed by applying H3.