let A be non empty set ; :: thesis: for O being Ordinal holds ConsecutiveSet A,(succ O) = new_set (ConsecutiveSet A,O)
let O be Ordinal; :: thesis: ConsecutiveSet A,(succ O) = new_set (ConsecutiveSet A,O)
deffunc H₁( Ordinal, set ) -> set = new_set $2;
deffunc H₂( Ordinal, T-Sequence) -> set = union (rng $2);
deffunc H₃( Ordinal) -> set = ConsecutiveSet 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) = H₁(C,L0 . C) ) & ( for C being Ordinal st C in succ O & C <> {} & C is limit_ordinal holds
L0 . C = H₂(C,L0 | C) ) ) ) by Def13;
for O being Ordinal holds H₃( succ O) = H₁(O,H₃(O)) from ORDINAL2:sch 9(A1);
hence ConsecutiveSet A,(succ O) = new_set (ConsecutiveSet A,O) ; :: thesis: verum