Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param explicit_Field_minus^{explicit_Field_minus} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι
Param ReplSep2^ReplSep2 : ι → (ι → ι) → (ι → ι → ο) → CT2 ι
Param True^True : ο
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Definition False^False := ∀ x0 : ο . x0
Theorem 801dc.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ι → ι → ι . (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x7 x8 ∈ x0) ⟶ x1 ∈ x0 ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x7 x8 ∈ x0) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x4 x7 (x4 x8 x9) = x4 (x4 x7 x8) x9) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x7 x8 = x4 x8 x7) ⟶ x2 ∈ x0 ⟶ (∀ x7 . x7 ∈ x0 ⟶ (x7 = x1 ⟶ ∀ x8 : ο . x8) ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ x0) (x4 x7 x9 = x2) ⟶ x8) ⟶ x8) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x4 x7 (x3 x8 x9) = x3 (x4 x7 x8) (x4 x7 x9)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ x0) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x4 (x3 x7 x8) x9 = x3 (x4 x7 x9) (x4 x8 x9)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 (x3 x7 x8) = x3 (explicit_Field_minus x0 x1 x2 x3 x4 x7) (explicit_Field_minus x0 x1 x2 x3 x4 x8)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 (explicit_Field_minus x0 x1 x2 x3 x4 x7) x8 = explicit_Field_minus x0 x1 x2 x3 x4 (x4 x7 x8)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x7 (explicit_Field_minus x0 x1 x2 x3 x4 x8) = explicit_Field_minus x0 x1 x2 x3 x4 (x4 x7 x8)) ⟶ (∀ x7 . x7 ∈ ReplSep2 x0 (λ x8 . x0) (λ x8 x9 . True) x6 ⟶ prim0 (λ x8 . and (x8 ∈ x0) (∀ x9 : ο . (∀ x10 . and (x10 ∈ x0) (x7 = x6 x8 x10) ⟶ x9) ⟶ x9)) ∈ x0) ⟶ (∀ x7 . x7 ∈ ReplSep2 x0 (λ x8 . x0) (λ x8 x9 . True) x6 ⟶ prim0 (λ x8 . and (x8 ∈ x0) (x7 = x6 (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x7 = x6 x10 x12) ⟶ x11) ⟶ x11))) x8)) ∈ x0) ⟶ (∀ x7 . x7 ∈ ReplSep2 x0 (λ x8 . x0) (λ x8 x9 . True) x6 ⟶ ∀ x8 . x8 ∈ ReplSep2 x0 (λ x9 . x0) (λ x9 x10 . True) x6 ⟶ prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x7 = x6 x10 x12) ⟶ x11) ⟶ x11)) = prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x8 = x6 x10 x12) ⟶ x11) ⟶ x11)) ⟶ prim0 (λ x10 . and (x10 ∈ x0) (x7 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x7 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10)) = prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x8 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10)) ⟶ x7 = x8) ⟶ (∀ x7 . x7 ∈ ReplSep2 x0 (λ x8 . x0) (λ x8 x9 . True) x6 ⟶ ∀ x8 . x8 ∈ ReplSep2 x0 (λ x9 . x0) (λ x9 x10 . True) x6 ⟶ x6 (x3 (x4 (prim0 (λ x9 . and (x9 ∈ x0) (∀ x10 : ο . (∀ x11 . and (x11 ∈ x0) (x7 = x6 x9 x11) ⟶ x10) ⟶ x10))) (prim0 (λ x9 . and (x9 ∈ x0) (∀ x10 : ο . (∀ x11 . and (x11 ∈ x0) (x8 = x6 x9 x11) ⟶ x10) ⟶ x10)))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x9 . and (x9 ∈ x0) (x7 = x6 (prim0 (λ x11 . and (x11 ∈ x0) (∀ x12 : ο . (∀ x13 . and (x13 ∈ x0) (x7 = x6 x11 x13) ⟶ x12) ⟶ x12))) x9))) (prim0 (λ x9 . and (x9 ∈ x0) (x8 = x6 (prim0 (λ x11 . and (x11 ∈ x0) (∀ x12 : ο . (∀ x13 . and (x13 ∈ x0) (x8 = x6 x11 x13) ⟶ x12) ⟶ x12))) x9)))))) (x3 (x4 (prim0 (λ x9 . and (x9 ∈ x0) (∀ x10 : ο . (∀ x11 . and (x11 ∈ x0) (x7 = x6 x9 x11) ⟶ x10) ⟶ x10))) (prim0 (λ x9 . and (x9 ∈ x0) (x8 = x6 (prim0 (λ x11 . and (x11 ∈ x0) (∀ x12 : ο . (∀ x13 . and (x13 ∈ x0) (x8 = x6 x11 x13) ⟶ x12) ⟶ x12))) x9)))) (x4 (prim0 (λ x9 . and (x9 ∈ x0) (x7 = x6 (prim0 (λ x11 . and (x11 ∈ x0) (∀ x12 : ο . (∀ x13 . and (x13 ∈ x0) (x7 = x6 x11 x13) ⟶ x12) ⟶ x12))) x9))) (prim0 (λ x9 . and (x9 ∈ x0) (∀ x10 : ο . (∀ x11 . and (x11 ∈ x0) (x8 = x6 x9 x11) ⟶ x10) ⟶ x10))))) ∈ ReplSep2 x0 (λ x9 . x0) (λ x9 x10 . True) x6) ⟶ (∀ x7 . x7 ∈ ReplSep2 x0 (λ x8 . x0) (λ x8 x9 . True) x6 ⟶ ∀ x8 . x8 ∈ ReplSep2 x0 (λ x9 . x0) (λ x9 x10 . True) x6 ⟶ prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x6 (x3 (x4 (prim0 (λ x14 . and (x14 ∈ x0) (∀ x15 : ο . (∀ x16 . and (x16 ∈ x0) (x7 = x6 x14 x16) ⟶ x15) ⟶ x15))) (prim0 (λ x14 . and (x14 ∈ x0) (∀ x15 : ο . (∀ x16 . and (x16 ∈ x0) (x8 = x6 x14 x16) ⟶ x15) ⟶ x15)))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x14 . and (x14 ∈ x0) (x7 = x6 (prim0 (λ x16 . and (x16 ∈ x0) (∀ x17 : ο . (∀ x18 . and (x18 ∈ x0) (x7 = x6 x16 x18) ⟶ x17) ⟶ x17))) x14))) (prim0 (λ x14 . and (x14 ∈ x0) (x8 = x6 (prim0 (λ x16 . and (x16 ∈ x0) (∀ x17 : ο . (∀ x18 . and (x18 ∈ x0) (x8 = x6 x16 x18) ⟶ x17) ⟶ x17))) x14)))))) (x3 (x4 (prim0 (λ x14 . and (x14 ∈ x0) (∀ x15 : ο . (∀ x16 . and (x16 ∈ x0) (x7 = x6 x14 x16) ⟶ x15) ⟶ x15))) (prim0 (λ x14 . and (x14 ∈ x0) (x8 = x6 (prim0 (λ x16 . and (x16 ∈ x0) (∀ x17 : ο . (∀ x18 . and (x18 ∈ x0) (x8 = x6 x16 x18) ⟶ x17) ⟶ x17))) x14)))) (x4 (prim0 (λ x14 . and (x14 ∈ x0) (x7 = x6 (prim0 (λ x16 . and (x16 ∈ x0) (∀ x17 : ο . (∀ x18 . and (x18 ∈ x0) (x7 = x6 x16 x18) ⟶ x17) ⟶ x17))) x14))) (prim0 (λ x14 . and (x14 ∈ x0) (∀ x15 : ο . (∀ x16 . and (x16 ∈ x0) (x8 = x6 x14 x16) ⟶ x15) ⟶ x15))))) = x6 x10 x12) ⟶ x11) ⟶ x11)) = x3 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x7 = x6 x10 x12) ⟶ x11) ⟶ x11))) (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x8 = x6 x10 x12) ⟶ x11) ⟶ x11)))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (x7 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x7 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x8 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10)))))) ⟶ (∀ x7 . x7 ∈ ReplSep2 x0 (λ x8 . x0) (λ x8 x9 . True) x6 ⟶ ∀ x8 . x8 ∈ ReplSep2 x0 (λ x9 . x0) (λ x9 x10 . True) x6 ⟶ prim0 (λ x10 . and (x10 ∈ x0) (x6 (x3 (x4 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x7 = x6 x12 x14) ⟶ x13) ⟶ x13))) (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x8 = x6 x12 x14) ⟶ x13) ⟶ x13)))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x12 . and (x12 ∈ x0) (x7 = x6 (prim0 (λ x14 . and (x14 ∈ x0) (∀ x15 : ο . (∀ x16 . and (x16 ∈ x0) (x7 = x6 x14 x16) ⟶ x15) ⟶ x15))) x12))) (prim0 (λ x12 . and (x12 ∈ x0) (x8 = x6 (prim0 (λ x14 . and (x14 ∈ x0) (∀ x15 : ο . (∀ x16 . and (x16 ∈ x0) (x8 = x6 x14 x16) ⟶ x15) ⟶ x15))) x12)))))) (x3 (x4 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x7 = x6 x12 x14) ⟶ x13) ⟶ x13))) (prim0 (λ x12 . and (x12 ∈ x0) (x8 = x6 (prim0 (λ x14 . and (x14 ∈ x0) (∀ x15 : ο . (∀ x16 . and (x16 ∈ x0) (x8 = x6 x14 x16) ⟶ x15) ⟶ x15))) x12)))) (x4 (prim0 (λ x12 . and (x12 ∈ x0) (x7 = x6 (prim0 (λ x14 . and (x14 ∈ x0) (∀ x15 : ο . (∀ x16 . and (x16 ∈ x0) (x7 = x6 x14 x16) ⟶ x15) ⟶ x15))) x12))) (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x8 = x6 x12 x14) ⟶ x13) ⟶ x13))))) = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x6 (x3 (x4 (prim0 (λ x16 . and (x16 ∈ x0) (∀ x17 : ο . (∀ x18 . and (x18 ∈ x0) (x7 = x6 x16 x18) ⟶ x17) ⟶ x17))) (prim0 (λ x16 . and (x16 ∈ x0) (∀ x17 : ο . (∀ x18 . and (x18 ∈ x0) (x8 = x6 x16 x18) ⟶ x17) ⟶ x17)))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x16 . and (x16 ∈ x0) (x7 = x6 (prim0 (λ x18 . and (x18 ∈ x0) (∀ x19 : ο . (∀ x20 . and (x20 ∈ x0) (x7 = x6 x18 x20) ⟶ x19) ⟶ x19))) x16))) (prim0 (λ x16 . and (x16 ∈ x0) (x8 = x6 (prim0 (λ x18 . and (x18 ∈ x0) (∀ x19 : ο . (∀ x20 . and (x20 ∈ x0) (x8 = x6 x18 x20) ⟶ x19) ⟶ x19))) x16)))))) (x3 (x4 (prim0 (λ x16 . and (x16 ∈ x0) (∀ x17 : ο . (∀ x18 . and (x18 ∈ x0) (x7 = x6 x16 x18) ⟶ x17) ⟶ x17))) (prim0 (λ x16 . and (x16 ∈ x0) (x8 = x6 (prim0 (λ x18 . and (x18 ∈ x0) (∀ x19 : ο . (∀ x20 . and (x20 ∈ x0) (x8 = x6 x18 x20) ⟶ x19) ⟶ x19))) x16)))) (x4 (prim0 (λ x16 . and (x16 ∈ x0) (x7 = x6 (prim0 (λ x18 . and (x18 ∈ x0) (∀ x19 : ο . (∀ x20 . and (x20 ∈ x0) (x7 = x6 x18 x20) ⟶ x19) ⟶ x19))) x16))) (prim0 (λ x16 . and (x16 ∈ x0) (∀ x17 : ο . (∀ x18 . and (x18 ∈ x0) (x8 = x6 x16 x18) ⟶ x17) ⟶ x17))))) = x6 x12 x14) ⟶ x13) ⟶ x13))) x10)) = x3 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x7 = x6 x10 x12) ⟶ x11) ⟶ x11))) (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x8 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10)))) (x4 (prim0 (λ x10 . and (x10 ∈ x0) (x7 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x7 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x8 = x6 x10 x12) ⟶ x11) ⟶ x11))))) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x6 x7 x8 ∈ ReplSep2 x0 (λ x9 . x0) (λ x9 x10 . True) x6) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ prim0 (λ x10 . and (x10 ∈ x0) (∀ x11 : ο . (∀ x12 . and (x12 ∈ x0) (x6 x7 x8 = x6 x10 x12) ⟶ x11) ⟶ x11)) = x7) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ prim0 (λ x10 . and (x10 ∈ x0) (x6 x7 x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∀ x13 : ο . (∀ x14 . and (x14 ∈ x0) (x6 x7 x8 = x6 x12 x14) ⟶ x13) ⟶ x13))) x10)) = x8) ⟶ x6 x1 x1 ∈ ReplSep2 x0 (λ x7 . x0) (λ x7 x8 . True) x6 ⟶ x6 x2 x1 ∈ ReplSep2 x0 (λ x7 . x0) (λ x7 x8 . True) x6 ⟶ (∀ x7 . x7 ∈ x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 (explicit_Field_minus x0 x1 x2 x3 x4 x7) = x7) ⟶ (∀ x7 . x7 ∈ x0 ⟶ x3 (explicit_Field_minus x0 x1 x2 x3 x4 x7) x7 = x1) ⟶ (∀ x7 . x7 ∈ x0 ⟶ x4 x1 x7 = x1) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 (x4 x7 x7) (x4 x8 x8) = x1 ⟶ and (x7 = x1) (x8 = x1)) ⟶ ∀ x7 . x7 ∈ ReplSep2 x0 (λ x8 . x0) (λ x8 x9 . True) x6 ⟶ (x7 = x6 x1 x1 ⟶ ∀ x8 : ο . x8) ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ ReplSep2 x0 (λ x10 . x0) (λ x10 x11 . True) x6) (x6 (x3 (x4 (prim0 (λ x11 . and (x11 ∈ x0) (∀ x12 : ο . (∀ x13 . and (x13 ∈ x0) (x7 = x6 x11 x13) ⟶ x12) ⟶ x12))) (prim0 (λ x11 . and (x11 ∈ x0) (∀ x12 : ο . (∀ x13 . and (x13 ∈ x0) (x9 = x6 x11 x13) ⟶ x12) ⟶ x12)))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x11 . and (x11 ∈ x0) (x7 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∀ x14 : ο . (∀ x15 . and (x15 ∈ x0) (x7 = x6 x13 x15) ⟶ x14) ⟶ x14))) x11))) (prim0 (λ x11 . and (x11 ∈ x0) (x9 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∀ x14 : ο . (∀ x15 . and (x15 ∈ x0) (x9 = x6 x13 x15) ⟶ x14) ⟶ x14))) x11)))))) (x3 (x4 (prim0 (λ x11 . and (x11 ∈ x0) (∀ x12 : ο . (∀ x13 . and (x13 ∈ x0) (x7 = x6 x11 x13) ⟶ x12) ⟶ x12))) (prim0 (λ x11 . and (x11 ∈ x0) (x9 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∀ x14 : ο . (∀ x15 . and (x15 ∈ x0) (x9 = x6 x13 x15) ⟶ x14) ⟶ x14))) x11)))) (x4 (prim0 (λ x11 . and (x11 ∈ x0) (x7 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∀ x14 : ο . (∀ x15 . and (x15 ∈ x0) (x7 = x6 x13 x15) ⟶ x14) ⟶ x14))) x11))) (prim0 (λ x11 . and (x11 ∈ x0) (∀ x12 : ο . (∀ x13 . and (x13 ∈ x0) (x9 = x6 x11 x13) ⟶ x12) ⟶ x12))))) = x6 x2 x1) ⟶ x8) ⟶ x8 (proof)