Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι → ι be given.

Let x3 of type ι → ι be given.

Let x4 of type ι → ι be given.

Let x5 of type ι → ι be given.

Let x6 of type ι → ι → ο be given.

Let x7 of type ι → ι → ο be given.

Let x8 of type ι → ι → ο be given.

Let x9 of type ι → ι → ο be given.

Assume H0: dd5e6.. x0 x2 x4 x6 x8 = dd5e6.. x1 x3 x5 x7 x9.

Claim L1: ...

...

Claim L2: ...

...

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

The subproof is completed by applying L2.

Let x10 of type ι be given.

Assume H3: prim1 x10 x0.

Apply unknownprop_5c0dfa81d135413630d40198f380b99880d733daeee8e129b4a5d28d7d207083 with x0, x2, x4, x6, x8, x10, λ x11 x12 . x12 = x3 x10 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: prim1 x10 x1

Apply L2 with λ x11 x12 . prim1 x10 x11.

The subproof is completed by applying H3.

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

Let x11 of type ι → ι → ο be given.

Apply unknownprop_5c0dfa81d135413630d40198f380b99880d733daeee8e129b4a5d28d7d207083 with x1, x3, x5, x7, x9, x10, λ x12 x13 . x11 x13 x12.

The subproof is completed by applying L4.

Let x10 of type ι be given.

Assume H3: prim1 x10 x0.

Apply unknownprop_9a2968cd6fb30108e6f9e7f55ebbd94ee83065a46958e63af2137d297ff71f1e with x0, x2, x4, x6, x8, x10, λ x11 x12 . x12 = x5 x10 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: prim1 x10 x1

Apply L2 with λ x11 x12 . prim1 x10 x11.

The subproof is completed by applying H3.

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

Let x11 of type ι → ι → ο be given.

Apply unknownprop_9a2968cd6fb30108e6f9e7f55ebbd94ee83065a46958e63af2137d297ff71f1e with x1, x3, x5, x7, x9, x10, λ x12 x13 . x11 x13 x12.

The subproof is completed by applying L4.

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_5cb8388f483d51e46bdfc58663301ae218a26e1890aaa5ae374d943cb2c0b71e with x0, x2, x4, x6, x8, x10, x11, λ x12 x13 : ο . 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 . 2b2e3.. (f482f.. x13 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x10 x11 = x7 x10 x11.

Let x12 of type ο → ο → ο be given.

Apply unknownprop_5cb8388f483d51e46bdfc58663301ae218a26e1890aaa5ae374d943cb2c0b71e with x1, x3, x5, x7, x9, x10, x11, λ x13 x14 : ο . 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_01493f66cd210cbabae1129d96b075b27bb1d5f8be40aad66910afee0cc11964 with x0, x2, x4, x6, x8, x10, x11, λ x12 x13 : ο . x13 = x9 x10 x11 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: ...

...

Claim L6: prim1 x11 x1

...

Apply H0 with λ x12 x13 . 2b2e3.. (f482f.. x13 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x10 x11 = x9 x10 x11.

Let x12 of type ο → ο → ο be given.

Apply unknownprop_01493f66cd210cbabae1129d96b075b27bb1d5f8be40aad66910afee0cc11964 with x1, x3, x5, x7, x9, x10, x11, λ x13 x14 : ο . x12 x14 x13 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

■

Proofgold Proof