Proofgold Signed Transaction

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 Sep^Sep : ι → (ι → ο) → ι
Param explicit_Field^{explicit_Field} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Param explicit_Reals^{explicit_Reals} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Param explicit_Complex^{explicit_Complex} : ι → (ι → ι) → (ι → ι) → ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known explicit_Complex_I^{explicit_Complex_I} : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 x5 . ∀ x6 x7 : ι → ι → ι . explicit_Field x0 x3 x4 x6 x7 ⟶ (∃ x8 : ι → ι → ο . explicit_Reals {x10 ∈ x0|x1 x10 = x10} x3 x4 x6 x7 x8) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x2 x8 ∈ {x9 ∈ x0|x1 x9 = x9}) ⟶ x5 ∈ x0 ⟶ (∀ x8 . x8 ∈ x0 ⟶ x1 x8 ∈ x0) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x2 x8 ∈ x0) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x8 = x6 (x1 x8) (x7 x5 (x2 x8))) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x1 x8 = x1 x9 ⟶ x2 x8 = x2 x9 ⟶ x8 = x9) ⟶ x6 (x7 x5 x5) x4 = x3 ⟶ explicit_Complex x0 x1 x2 x3 x4 x5 x6 x7
Known and6I^and6I : ∀ x0 x1 x2 x3 x4 x5 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ and (and (and (and (and x0 x1) x2) x3) x4) x5
Theorem 256ef.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ι → ι → ι . ∀ x7 . (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x6 x8 x9 = x6 x10 x11 ⟶ and (x8 = x10) (x9 = x11)) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x3 x8 x9 ∈ x0) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x3 x8 x9 = x3 x9 x8) ⟶ x1 ∈ x0 ⟶ (∀ x8 . x8 ∈ x0 ⟶ x3 x1 x8 = x8) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x4 x8 x9 ∈ x0) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x4 x8 x9 = x4 x9 x8) ⟶ x2 ∈ x0 ⟶ (∀ x8 . x8 ∈ x0 ⟶ x4 x2 x8 = x8) ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x2 ∈ x0 ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x6 x8 x9 ∈ x7) ⟶ (∀ x8 . x8 ∈ x7 ⟶ ∀ x9 : ι → ο . (∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x8 = x6 x10 x11 ⟶ x9 (x6 x10 x11)) ⟶ x9 x8) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ prim0 (λ x11 . and (x11 ∈ x0) (∃ x12 . and (x12 ∈ x0) (x6 x8 x9 = x6 x11 x12))) = x8) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ prim0 (λ x11 . and (x11 ∈ x0) (x6 x8 x9 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x6 x8 x9 = x6 x13 x14)))) x11)) = x9) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x6 x8 x1 ∈ {x9 ∈ x7|x6 (prim0 (λ x11 . and (x11 ∈ x0) (∃ x12 . and (x12 ∈ x0) (x9 = x6 x11 x12)))) x1 = x9}) ⟶ (∀ x8 . x8 ∈ x7 ⟶ prim0 (λ x9 . and (x9 ∈ x0) (∃ x10 . and (x10 ∈ x0) (x8 = x6 x9 x10))) ∈ x0) ⟶ (∀ x8 . x8 ∈ x7 ⟶ prim0 (λ x9 . and (x9 ∈ x0) (x8 = x6 (prim0 (λ x11 . and (x11 ∈ x0) (∃ x12 . and (x12 ∈ x0) (x8 = x6 x11 x12)))) x9)) ∈ x0) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x6 (x3 (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x6 x8 x9 = x6 x13 x14)))) (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x6 x10 x11 = x6 x13 x14))))) (x3 (prim0 (λ x13 . and (x13 ∈ x0) (x6 x8 x9 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∃ x16 . and (x16 ∈ x0) (x6 x8 x9 = x6 x15 x16)))) x13))) (prim0 (λ x13 . and (x13 ∈ x0) (x6 x10 x11 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∃ x16 . and (x16 ∈ x0) (x6 x10 x11 = x6 x15 x16)))) x13)))) = x6 (x3 x8 x10) (x3 x9 x11)) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x6 (x3 (x4 (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x6 x8 x9 = x6 x13 x14)))) (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x6 x10 x11 = x6 x13 x14))))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x13 . and (x13 ∈ x0) (x6 x8 x9 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∃ x16 . and (x16 ∈ x0) (x6 x8 x9 = x6 x15 x16)))) x13))) (prim0 (λ x13 . and (x13 ∈ x0) (x6 x10 x11 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∃ x16 . and (x16 ∈ x0) (x6 x10 x11 = x6 x15 x16)))) x13)))))) (x3 (x4 (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x6 x8 x9 = x6 x13 x14)))) (prim0 (λ x13 . and (x13 ∈ x0) (x6 x10 x11 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∃ x16 . and (x16 ∈ x0) (x6 x10 x11 = x6 x15 x16)))) x13)))) (x4 (prim0 (λ x13 . and (x13 ∈ x0) (x6 x8 x9 = x6 (prim0 (λ x15 . and (x15 ∈ x0) (∃ x16 . and (x16 ∈ x0) (x6 x8 x9 = x6 x15 x16)))) x13))) (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x6 x10 x11 = x6 x13 x14)))))) = x6 (x3 (x4 x8 x10) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 x9 x11))) (x3 (x4 x8 x11) (x4 x9 x10))) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x3 (explicit_Field_minus x0 x1 x2 x3 x4 x8) x8 = x1) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x3 x8 (explicit_Field_minus x0 x1 x2 x3 x4 x8) = x1) ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x1 = x1 ⟶ (∀ x8 . x8 ∈ x0 ⟶ x4 x1 x8 = x1) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x4 x8 x1 = x1) ⟶ explicit_Field x7 (x6 x1 x1) (x6 x2 x1) (λ x8 x9 . x6 (x3 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x9 = x6 x10 x11))))) (x3 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x9 = x6 x12 x13)))) x10))))) (λ x8 x9 . x6 (x3 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x9 = x6 x10 x11))))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x9 = x6 x12 x13)))) x10)))))) (x3 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x9 = x6 x12 x13)))) x10)))) (x4 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x9 = x6 x10 x11))))))) ⟶ explicit_Reals {x8 ∈ x7|x6 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) x1 = x8} (x6 x1 x1) (x6 x2 x1) (λ x8 x9 . x6 (x3 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x9 = x6 x10 x11))))) (x3 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x9 = x6 x12 x13)))) x10))))) (λ x8 x9 . x6 (x3 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x9 = x6 x10 x11))))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x9 = x6 x12 x13)))) x10)))))) (x3 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x9 = x6 x12 x13)))) x10)))) (x4 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x9 = x6 x10 x11))))))) (λ x8 x9 . x5 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x9 = x6 x10 x11))))) ⟶ and (explicit_Complex x7 (λ x8 . x6 (prim0 (λ x9 . and (x9 ∈ x0) (∃ x10 . and (x10 ∈ x0) (x8 = x6 x9 x10)))) x1) (λ x8 . x6 (prim0 (λ x9 . and (x9 ∈ x0) (x8 = x6 (prim0 (λ x11 . and (x11 ∈ x0) (∃ x12 . and (x12 ∈ x0) (x8 = x6 x11 x12)))) x9))) x1) (x6 x1 x1) (x6 x2 x1) (x6 x1 x2) (λ x8 x9 . x6 (x3 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x9 = x6 x10 x11))))) (x3 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x9 = x6 x12 x13)))) x10))))) (λ x8 x9 . x6 (x3 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x9 = x6 x10 x11))))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x9 = x6 x12 x13)))) x10)))))) (x3 (x4 (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11)))) (prim0 (λ x10 . and (x10 ∈ x0) (x9 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x9 = x6 x12 x13)))) x10)))) (x4 (prim0 (λ x10 . and (x10 ∈ x0) (x8 = x6 (prim0 (λ x12 . and (x12 ∈ x0) (∃ x13 . and (x13 ∈ x0) (x8 = x6 x12 x13)))) x10))) (prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x9 = x6 x10 x11)))))))) ((∀ x8 . x8 ∈ x0 ⟶ x6 x8 x1 = x8) ⟶ and (and (and (and (and (x0 ⊆ x7) (∀ x8 . x8 ∈ x0 ⟶ prim0 (λ x10 . and (x10 ∈ x0) (∃ x11 . and (x11 ∈ x0) (x8 = x6 x10 x11))) = x8)) (x6 x1 x1 = x1)) (x6 x2 x1 = x2)) (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x6 (x3 (prim0 (λ x11 . and (x11 ∈ x0) (∃ x12 . and (x12 ∈ x0) (x8 = x6 x11 x12)))) (prim0 (λ x11 . and (x11 ∈ x0) (∃ x12 . and (x12 ∈ x0) (x9 = x6 x11 x12))))) (x3 (prim0 (λ x11 . and (x11 ∈ x0) (x8 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x8 = x6 x13 x14)))) x11))) (prim0 (λ x11 . and (x11 ∈ x0) (x9 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x9 = x6 x13 x14)))) x11)))) = x3 x8 x9)) (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x6 (x3 (x4 (prim0 (λ x11 . and (x11 ∈ x0) (∃ x12 . and (x12 ∈ x0) (x8 = x6 x11 x12)))) (prim0 (λ x11 . and (x11 ∈ x0) (∃ x12 . and (x12 ∈ x0) (x9 = x6 x11 x12))))) (explicit_Field_minus x0 x1 x2 x3 x4 (x4 (prim0 (λ x11 . and (x11 ∈ x0) (x8 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x8 = x6 x13 x14)))) x11))) (prim0 (λ x11 . and (x11 ∈ x0) (x9 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x9 = x6 x13 x14)))) x11)))))) (x3 (x4 (prim0 (λ x11 . and (x11 ∈ x0) (∃ x12 . and (x12 ∈ x0) (x8 = x6 x11 x12)))) (prim0 (λ x11 . and (x11 ∈ x0) (x9 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x9 = x6 x13 x14)))) x11)))) (x4 (prim0 (λ x11 . and (x11 ∈ x0) (x8 = x6 (prim0 (λ x13 . and (x13 ∈ x0) (∃ x14 . and (x14 ∈ x0) (x8 = x6 x13 x14)))) x11))) (prim0 (λ x11 . and (x11 ∈ x0) (∃ x12 . and (x12 ∈ x0) (x9 = x6 x11 x12)))))) = x4 x8 x9))