let A be non empty set ; :: thesis: for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))

let L be lower-bounded LATTICE; :: thesis: for O being Ordinal
for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))

let O be Ordinal; :: thesis: for d being BiFunction of A,L
for q being QuadrSeq of d holds ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))

deffunc H1( Ordinal, Sequence) -> set = union (rng \$2);
let d be BiFunction of A,L; :: thesis: for q being QuadrSeq of d holds ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))
let q be QuadrSeq of d; :: thesis: ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O)))
deffunc H2( Ordinal, set ) -> BiFunction of (new_set (ConsecutiveSet (A,\$1))),L = new_bi_fun ((BiFun (\$2,(ConsecutiveSet (A,\$1)),L)),(Quadr (q,\$1)));
deffunc H3( Ordinal) -> set = ConsecutiveDelta (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 Def16;
for O being Ordinal holds H3( succ O) = H2(O,H3(O)) from hence ConsecutiveDelta (q,(succ O)) = new_bi_fun ((BiFun ((ConsecutiveDelta (q,O)),(ConsecutiveSet (A,O)),L)),(Quadr (q,O))) ; :: thesis: verum