deffunc H₁( Ordinal, set ) -> set = ( { u where u is Element of Tarski-Class X : ex v being Element of Tarski-Class X st
( v in $2 & u c= v ) } \/ { (bool v) where v is Element of Tarski-Class X : v in $2 } ) \/ ((bool $2) /\ (Tarski-Class X));
deffunc H₂( Ordinal, T-Sequence) -> set = (union (rng $2)) /\ (Tarski-Class X);
thus ( ex x being set ex L being T-Sequence st
( x = last L & dom L = succ A & L . {} = {X} & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = H₁(C,L . C) ) & ( for C being Ordinal st C in succ A & C <> {} & C is limit_ordinal holds
L . C = H₂(C,L | C) ) ) & ( for x1, x2 being set st ex L being T-Sequence st
( x1 = last L & dom L = succ A & L . {} = {X} & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = H₁(C,L . C) ) & ( for C being Ordinal st C in succ A & C <> {} & C is limit_ordinal holds
L . C = H₂(C,L | C) ) ) & ex L being T-Sequence st
( x2 = last L & dom L = succ A & L . {} = {X} & ( for C being Ordinal st succ C in succ A holds
L . (succ C) = H₁(C,L . C) ) & ( for C being Ordinal st C in succ A & C <> {} & C is limit_ordinal holds
L . C = H₂(C,L | C) ) ) holds
x1 = x2 ) ) from ORDINAL2:sch 7(); :: thesis: verum