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 unknownprop_8acfb80b309c166e5c3c41e4a1cc49c4ea05db3f03d215384dabecf7c22c27a2 with
y2,
λ x5 x6 . add_SNo y2 y3 = add_SNo x6 (28f8d.. y3),
λ x5 . x4 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply unknownprop_8acfb80b309c166e5c3c41e4a1cc49c4ea05db3f03d215384dabecf7c22c27a2 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 unknownprop_872273e895264b163d3a3b042c5d1abf262e26919401a643ccce2dcdcb6a14ef with
add_SNo (28f8d.. y2) (28f8d.. 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 unknownprop_2a9fa88c4206964d15bfbbc297f8b3b39425bd997c7d45b304d4d13c3943fd64 with
y5,
λ x10 x11 . 0 = add_SNo x11 (d634d.. y6),
λ x10 x11 . y9 (ad280.. (add_SNo (28f8d.. y5) (28f8d.. y6)) x10) (ad280.. (add_SNo (28f8d.. y5) (28f8d.. y6)) x11) leaving 3 subgoals.
The subproof is completed by applying H0.
Apply unknownprop_2a9fa88c4206964d15bfbbc297f8b3b39425bd997c7d45b304d4d13c3943fd64 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.