deffunc H₁( set , Sequence) -> set = union (rng $2);
let A be non empty set ; :: thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
let L be lower-bounded LATTICE; :: thesis: for d being BiFunction of A,L
for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
let d be BiFunction of A,L; :: thesis: for q being QuadrSeq of d
for O being Ordinal holds ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
let q be QuadrSeq of d; :: thesis: for O being Ordinal holds ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
let O be Ordinal; :: thesis: ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O)))
deffunc H₂( Ordinal, set ) -> BiFunction of (new_set2 (ConsecutiveSet2 (A,$1))),L = new_bi_fun2 ((BiFun ($2,(ConsecutiveSet2 (A,$1)),L)),(Quadr2 (q,$1)));
deffunc H₃( Ordinal) -> set = ConsecutiveDelta2 (q,$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 = 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 <> 0 & C is limit_ordinal holds
L0 . C = H₁(C,L0 | C) ) ) ) by Def7;
for O being Ordinal holds H₃( succ O) = H₂(O,H₃(O)) from ORDINAL2:sch 9(A1);
hence ConsecutiveDelta2 (q,(succ O)) = new_bi_fun2 ((BiFun ((ConsecutiveDelta2 (q,O)),(ConsecutiveSet2 (A,O)),L)),(Quadr2 (q,O))) ; :: thesis: verum