deffunc H₁( set , set ) -> set = (union { (Compound (s,$2)) where s is ofAtomicFormula Element of S : s is operational } ) \/ $2;
let IT1, IT2 be Function; :: thesis: ( dom IT1 = NAT & IT1 . 0 = AtomicTermsOf S & ( for n being Nat holds IT1 . (n + 1) = (union { (Compound (s,(IT1 . n))) where s is ofAtomicFormula Element of S : s is operational } ) \/ (IT1 . n) ) & dom IT2 = NAT & IT2 . 0 = AtomicTermsOf S & ( for n being Nat holds IT2 . (n + 1) = (union { (Compound (s,(IT2 . n))) where s is ofAtomicFormula Element of S : s is operational } ) \/ (IT2 . n) ) implies IT1 = IT2 )
assume A1: ( dom IT1 = NAT & IT1 . 0 = AtomicTermsOf S & ( for n being Nat holds IT1 . (n + 1) = H₁(n,IT1 . n) ) ) ; :: thesis: ( not dom IT2 = NAT or not IT2 . 0 = AtomicTermsOf S or ex n being Nat st not IT2 . (n + 1) = (union { (Compound (s,(IT2 . n))) where s is ofAtomicFormula Element of S : s is operational } ) \/ (IT2 . n) or IT1 = IT2 )
B1: dom IT1 = NAT by A1;
B2: IT1 . 0 = AtomicTermsOf S by A1;
B3: for n being Nat holds IT1 . (n + 1) = H₁(n,IT1 . n) by A1;
assume A2: ( dom IT2 = NAT & IT2 . 0 = AtomicTermsOf S & ( for n being Nat holds IT2 . (n + 1) = H₁(n,IT2 . n) ) ) ; :: thesis: IT1 = IT2
B4: dom IT2 = NAT by A2;
B5: IT2 . 0 = AtomicTermsOf S by A2;
B6: for n being Nat holds IT2 . (n + 1) = H₁(n,IT2 . n) by A2;
thus IT1 = IT2 from NAT_1:sch 15(B1, B2, B3, B4, B5, B6); :: thesis: verum