deffunc H₁( Ordinal, Sequence) -> set = union (rng $2);
deffunc H₂( Ordinal, set ) -> BiFunction of (new_set (ConsecutiveSet (A,$1))),L = new_bi_fun ((BiFun ($2,(ConsecutiveSet (A,$1)),L)),(Quadr (q,$1)));
( ex x being object ex L1 being Sequence st
( x = last L1 & dom L1 = succ O & L1 . 0 = d & ( for C being Ordinal st succ C in succ O holds
L1 . (succ C) = H₂(C,L1 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L1 . C = H₁(C,L1 | C) ) ) & ( for x1, x2 being set st ex L1 being Sequence st
( x1 = last L1 & dom L1 = succ O & L1 . 0 = d & ( for C being Ordinal st succ C in succ O holds
L1 . (succ C) = H₂(C,L1 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L1 . C = H₁(C,L1 | C) ) ) & ex L1 being Sequence st
( x2 = last L1 & dom L1 = succ O & L1 . 0 = d & ( for C being Ordinal st succ C in succ O holds
L1 . (succ C) = H₂(C,L1 . C) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L1 . C = H₁(C,L1 | C) ) ) holds
x1 = x2 ) ) from ORDINAL2:sch 7();
hence ( ex b₁ being set ex L0 being Sequence st
( b₁ = last L0 & dom L0 = succ O & L0 . 0 = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun ((BiFun ((L0 . C),(ConsecutiveSet (A,C)),L)),(Quadr (q,C))) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) & ( for b₁, b₂ being set st ex L0 being Sequence st
( b₁ = last L0 & dom L0 = succ O & L0 . 0 = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun ((BiFun ((L0 . C),(ConsecutiveSet (A,C)),L)),(Quadr (q,C))) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) & ex L0 being Sequence st
( b₂ = last L0 & dom L0 = succ O & L0 . 0 = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = new_bi_fun ((BiFun ((L0 . C),(ConsecutiveSet (A,C)),L)),(Quadr (q,C))) ) & ( for C being Ordinal st C in succ O & C <> 0 & C is limit_ordinal holds
L0 . C = union (rng (L0 | C)) ) ) holds
b₁ = b₂ ) ) ; :: thesis: