deffunc H₁( Ordinal, T-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)));
thus ( ex x being set ex L1 being T-Sequence st
( x = last L1 & dom L1 = succ O & L1 . {} = 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 <> {} & C is limit_ordinal holds
L1 . C = H₁(C,L1 | C) ) ) & ( for x1, x2 being set st ex L1 being T-Sequence st
( x1 = last L1 & dom L1 = succ O & L1 . {} = 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 <> {} & C is limit_ordinal holds
L1 . C = H₁(C,L1 | C) ) ) & ex L1 being T-Sequence st
( x2 = last L1 & dom L1 = succ O & L1 . {} = 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 <> {} & C is limit_ordinal holds
L1 . C = H₁(C,L1 | C) ) ) holds
x1 = x2 ) ) from ORDINAL2:sch 7(); :: thesis: verum