set B = card omega;
deffunc H₁( Ordinal, Sequence) -> Cardinal = card (sup $2);
deffunc H₂( Ordinal, set ) -> Cardinal = nextcard $2;
( ex x being object ex S being Sequence st
( x = last S & dom S = succ A & S . 0 = card omega & ( for C being Ordinal st succ C in succ A holds
S . (succ C) = H₂(C,S . C) ) & ( for C being Ordinal st C in succ A & C <> 0 & C is limit_ordinal holds
S . C = H₁(C,S | C) ) ) & ( for x1, x2 being set st ex S being Sequence st
( x1 = last S & dom S = succ A & S . 0 = card omega & ( for C being Ordinal st succ C in succ A holds
S . (succ C) = H₂(C,S . C) ) & ( for C being Ordinal st C in succ A & C <> 0 & C is limit_ordinal holds
S . C = H₁(C,S | C) ) ) & ex S being Sequence st
( x2 = last S & dom S = succ A & S . 0 = card omega & ( for C being Ordinal st succ C in succ A holds
S . (succ C) = H₂(C,S . C) ) & ( for C being Ordinal st C in succ A & C <> 0 & C is limit_ordinal holds
S . C = H₁(C,S | C) ) ) holds
x1 = x2 ) ) from ORDINAL2:sch 7();
hence ( ex b₁ being set ex S being Sequence st
( b₁ = last S & dom S = succ A & S . 0 = card omega & ( for B being Ordinal st succ B in succ A holds
S . (succ B) = nextcard (S . B) ) & ( for B being Ordinal st B in succ A & B <> 0 & B is limit_ordinal holds
S . B = card (sup (S | B)) ) ) & ( for b₁, b₂ being set st ex S being Sequence st
( b₁ = last S & dom S = succ A & S . 0 = card omega & ( for B being Ordinal st succ B in succ A holds
S . (succ B) = nextcard (S . B) ) & ( for B being Ordinal st B in succ A & B <> 0 & B is limit_ordinal holds
S . B = card (sup (S | B)) ) ) & ex S being Sequence st
( b₂ = last S & dom S = succ A & S . 0 = card omega & ( for B being Ordinal st succ B in succ A holds
S . (succ B) = nextcard (S . B) ) & ( for B being Ordinal st B in succ A & B <> 0 & B is limit_ordinal holds
S . B = card (sup (S | B)) ) ) holds
b₁ = b₂ ) ) ; :: thesis: verum