deffunc H₁( set , T-Sequence) -> set = union (rng $2);
deffunc H₂( Ordinal, set ) -> BiFunction of (new_set2 (ConsecutiveSet2 A,$1)),L = new_bi_fun2 (BiFun $2,(ConsecutiveSet2 A,$1),L),(Quadr2 q,$1);
thus ( ex x being set ex L0 being T-Sequence st
( x = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H₂(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H₁(C,L0 | C) ) ) & ( for x1, x2 being set st ex L0 being T-Sequence st
( x1 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H₂(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H₁(C,L0 | C) ) ) & ex L0 being T-Sequence st
( x2 = last L0 & dom L0 = succ O & L0 . {} = d & ( for C being Ordinal st succ C in succ O holds
L0 . (succ C) = H₂(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H₁(C,L0 | C) ) ) holds
x1 = x2 ) ) from ORDINAL2:sch 7(); :: thesis: verum