Let x0 of type ι be given.
Let x1 of type ι be given.
Claim L2: ∀ x4 : ι → ο . x4 y3 ⟶ x4 y2
Let x4 of type ι → ο be given.
Apply SNo_Re with
y2,
λ x5 x6 . add_SNo y2 y3 = add_SNo x6 (CSNo_Re y3),
λ x5 . x4 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply SNo_Re with
y3,
λ x5 x6 . add_SNo y2 y3 = add_SNo y2 x6 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x5 of type ι → ι → ο be given.
The subproof is completed by applying H3.
set y5 to be λ x5 . x4
Apply SNo_pair_0 with
add_SNo (CSNo_Re y2) (CSNo_Re y3),
λ x6 x7 . y5 x7 x6.
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 SNo_Im with
y5,
λ x10 x11 . 0 = add_SNo x11 (CSNo_Im y6),
λ x10 x11 . y9 (SNo_pair (add_SNo (CSNo_Re y5) (CSNo_Re y6)) x10) (SNo_pair (add_SNo (CSNo_Re y5) (CSNo_Re y6)) x11) leaving 3 subgoals.
The subproof is completed by applying H0.
Apply SNo_Im with
y6,
λ x10 x11 . 0 = add_SNo 0 x11 leaving 2 subgoals.
The subproof is completed by applying H1.
Let x10 of type ι → ι → ο be given.
Apply add_SNo_0L with
0,
λ x11 x12 . x10 x12 x11.
The subproof is completed by applying SNo_0.
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.
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.