deffunc H₁( set , Sequence) -> set = union (rng $2);
deffunc H₂( Ordinal, set ) -> set = new_set2 $2;
let A be non empty set ; :: thesis: for O being Ordinal holds ConsecutiveSet2 (A,(succ O)) = new_set2 (ConsecutiveSet2 (A,O))
let O be Ordinal; :: thesis: ConsecutiveSet2 (A,(succ O)) = new_set2 (ConsecutiveSet2 (A,O))
deffunc H₃( Ordinal) -> set = ConsecutiveSet2 (A,$1);
A1: for O being Ordinal
for It being object holds
( It = H₃(O) iff ex L0 being Sequence st
( It = last L0 & dom L0 = succ O & L0 . 0 = 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 <> 0 & C is limit_ordinal holds
L0 . C = H₁(C,L0 | C) ) ) ) by Def5;
for O being Ordinal holds H₃( succ O) = H₂(O,H₃(O)) from ORDINAL2:sch 9(A1);
hence ConsecutiveSet2 (A,(succ O)) = new_set2 (ConsecutiveSet2 (A,O)) ; :: thesis: verum