deffunc H₁( set , T-Sequence) -> set = union (rng $2);
deffunc H₂( Ordinal, set ) -> set = new_set2 $2;
let A be non empty set ; :: thesis: ConsecutiveSet2 (A,{}) = A
deffunc H₃( Ordinal) -> set = ConsecutiveSet2 (A,$1);
A1: for O being Ordinal
for It being set holds
( It = H₃(O) iff ex L0 being T-Sequence st
( It = last L0 & dom L0 = succ O & L0 . {} = A & ( 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) ) ) ) by Def6;
thus H₃( {} ) = A from ORDINAL2:sch 8(A1); :: thesis: verum