Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι → ι → ι be given.

Let x3 of type ι → ι → ι be given.

Let x4 of type ι → ι → ι be given.

Let x5 of type ι → ι → ι be given.

Let x6 of type ι → ι → ι be given.

Let x7 of type ι → ι → ι be given.

Let x8 of type ι be given.

Let x9 of type ι be given.

Assume H0: 71338.. x0 x2 x4 x6 x8 = 71338.. x1 x3 x5 x7 x9.

Claim L1: ...

...

Claim L2: ...

...

Apply and5I with x0 = x1, ∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x2 x10 x11 = x3 x10 x11, ∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x4 x10 x11 = x5 x10 x11, ∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 x10 x11 = x7 x10 x11, x8 = x9 leaving 5 subgoals.

The subproof is completed by applying L2.

Let x10 of type ι be given.

Assume H3: prim1 x10 x0.

Let x11 of type ι be given.

Assume H4: prim1 x11 x0.

Apply unknownprop_9b454a2a80798cd0556f866bae028da0e2e318c1ba057a70c632226789ea8a84 with x0, x2, x4, x6, x8, x10, x11, λ x12 x13 . x13 = x3 x10 x11 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x10 x1

Apply L2 with λ x12 x13 . prim1 x10 x12.

The subproof is completed by applying H3.

Claim L6: prim1 x11 x1

Apply L2 with λ x12 x13 . prim1 x11 x12.

The subproof is completed by applying H4.

Apply H0 with λ x12 x13 . e3162.. (f482f.. x13 (4ae4a.. 4a7ef..)) x10 x11 = x3 x10 x11.

Let x12 of type ι → ι → ο be given.

Apply unknownprop_9b454a2a80798cd0556f866bae028da0e2e318c1ba057a70c632226789ea8a84 with x1, x3, x5, x7, x9, x10, x11, λ x13 x14 . x12 x14 x13 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Let x10 of type ι be given.

Assume H3: prim1 x10 x0.

Let x11 of type ι be given.

Assume H4: prim1 x11 x0.

Apply unknownprop_f31aaef5c858e3702eb2359462a8937983338781aa537e061f9d5fdc0f57e20d with x0, x2, x4, x6, x8, x10, x11, λ x12 x13 . x13 = x5 x10 x11 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x10 x1

Apply L2 with λ x12 x13 . prim1 x10 x12.

The subproof is completed by applying H3.

Claim L6: prim1 x11 x1

Apply L2 with λ x12 x13 . prim1 x11 x12.

The subproof is completed by applying H4.

Apply H0 with λ x12 x13 . e3162.. (f482f.. x13 (4ae4a.. (4ae4a.. 4a7ef..))) x10 x11 = x5 x10 x11.

Let x12 of type ι → ι → ο be given.

Apply unknownprop_f31aaef5c858e3702eb2359462a8937983338781aa537e061f9d5fdc0f57e20d with x1, x3, x5, x7, x9, x10, x11, λ x13 x14 . x12 x14 x13 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Let x10 of type ι be given.

Assume H3: prim1 x10 x0.

Let x11 of type ι be given.

Assume H4: prim1 x11 x0.

Apply unknownprop_e54ba49d9a3e3d8a71364e79ce9fdb2e0cc0dda16726fbe8dee22a1930324832 with x0, x2, x4, x6, x8, x10, x11, λ x12 x13 . x13 = x7 x10 x11 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x10 x1

Apply L2 with λ x12 x13 . prim1 x10 x12.

The subproof is completed by applying H3.

Claim L6: prim1 x11 x1

Apply L2 with λ x12 x13 . prim1 x11 x12.

The subproof is completed by applying H4.

Apply H0 with λ x12 x13 . e3162.. (f482f.. x13 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x10 x11 = x7 x10 x11.

Let x12 of type ι → ι → ο be given.

Apply unknownprop_e54ba49d9a3e3d8a71364e79ce9fdb2e0cc0dda16726fbe8dee22a1930324832 with x1, x3, x5, x7, x9, x10, x11, λ x13 x14 . x12 x14 x13 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Apply unknownprop_7fa7a5a74cf161314fac31f56afe4de41cd061d38975b981f056e284178787d5 with x0, x2, x4, x6, x8, λ x10 x11 . x11 = x9.

Apply H0 with λ x10 x11 . f482f.. ... ... = ....

...

■

Proofgold Proof