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 H₁( Ordinal, set ) -> BiFunction of (new_set2 (ConsecutiveSet2 A,$1)),L = new_bi_fun2 (BiFun $2,(ConsecutiveSet2 A,$1),L),(Quadr2 q,$1);
deffunc H₂( set , T-Sequence) -> set = union (rng $2);
deffunc H₃( Ordinal) -> set = ConsecutiveDelta2 q,$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 . {} = d & ( 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 Def8;
for O being Ordinal holds H₃( succ O) = H₁(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