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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ιιι be given.
Let x3 of type ιιι be given.
Apply unknownprop_90e0281ca50dd9937c4dc4598e4b18b763eeb528d21427cfd7d571699b566522 with λ x4 . λ x5 : ι → ι → ι . 93c99.. (987b2.. x0 x2) (λ x6 . λ x7 : ι → ι → ι . and (and (987b2.. x0 x2 = 987b2.. x6 x5) (4f2b4.. (987b2.. x6 x5))) (Subq x6 x4)), x1, x3, λ x4 x5 : ο . x5 = and (and (987b2.. x0 x2 = 987b2.. x0 x3) (4f2b4.. (987b2.. x0 x3))) (Subq x0 x1) leaving 2 subgoals.
Let x4 of type ιιι be given.
Assume H0: ∀ x5 . prim1 x5 x1∀ x6 . prim1 x6 x1x3 x5 x6 = x4 x5 x6.
Apply unknownprop_2c910a90451492bce41b148527a925f3c473a4798adb2a8da5328ebb68c7247f with x0, x1, x2, x3, λ x5 x6 : ο . 93c99.. (987b2.. x0 x2) (λ x7 . λ x8 : ι → ι → ι . and (and (987b2.. x0 x2 = 987b2.. x7 x4) (4f2b4.. (987b2.. x7 x4))) (Subq x7 x1)) = x6.
Apply unknownprop_2c910a90451492bce41b148527a925f3c473a4798adb2a8da5328ebb68c7247f with x0, x1, x2, x4, λ x5 x6 : ο . x6 = and (and (987b2.. x0 x2 = 987b2.. x0 x3) (4f2b4.. (987b2.. x0 x3))) (Subq x0 x1).
Claim L1: ...
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Apply prop_ext_2 with and (and (987b2.. x0 x2 = 987b2.. x0 x4) (4f2b4.. (987b2.. x0 x4))) (Subq x0 x1), and (and (987b2.. x0 x2 = 987b2.. x0 x3) (4f2b4.. (987b2.. x0 x3))) (Subq x0 x1) leaving 2 subgoals.
Assume H2: and (and (987b2.. x0 x2 = 987b2.. x0 x4) (4f2b4.. (987b2.. x0 x4))) (Subq x0 x1).
Apply H2 with and (and (987b2.. x0 x2 = 987b2.. x0 x3) (4f2b4.. (987b2.. x0 x3))) (Subq x0 x1).
Assume H3: and (987b2.. x0 x2 = 987b2.. x0 x4) (4f2b4.. (987b2.. x0 x4)).
Assume H4: Subq x0 x1.
Apply L1 with λ x5 x6 . and (and (987b2.. x0 x2 = x6) (4f2b4.. x6)) (Subq x0 x1) leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
Assume H2: and (and (987b2.. x0 x2 = 987b2.. x0 x3) (4f2b4.. (987b2.. x0 x3))) (Subq x0 x1).
Apply H2 with and (and (987b2.. x0 x2 = 987b2.. x0 x4) (4f2b4.. (987b2.. x0 x4))) (Subq x0 x1).
Assume H3: and (987b2.. x0 x2 = 987b2.. x0 x3) (4f2b4.. (987b2.. x0 x3)).
Assume H4: Subq x0 x1.
Apply L1 with λ x5 x6 . and (and (987b2.. x0 ... = ...) ...) ... leaving 2 subgoals.
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