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 d c= ConsecutiveDelta2 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 d c= ConsecutiveDelta2 q,O
let d be BiFunction of A,L; :: thesis: for q being QuadrSeq of d
for O being Ordinal holds d c= ConsecutiveDelta2 q,O
let q be QuadrSeq of d; :: thesis: for O being Ordinal holds d c= ConsecutiveDelta2 q,O
let O be Ordinal; :: thesis: d c= ConsecutiveDelta2 q,O
defpred S₁[ Ordinal] means d c= ConsecutiveDelta2 q,$1;
A1: for O1 being Ordinal st S₁[O1] holds
S₁[ succ O1]

proof

let O1 be Ordinal; :: thesis: ( S₁[O1] implies S₁[ succ O1] )
ConsecutiveDelta2 q,(succ O1) = new_bi_fun2 (BiFun (ConsecutiveDelta2 q,O1),(ConsecutiveSet2 A,O1),L),(Quadr2 q,O1) by Th20
.= new_bi_fun2 (ConsecutiveDelta2 q,O1),(Quadr2 q,O1) by LATTICE5:def 16 ;
then A2: ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,(succ O1) by Th14;
assume d c= ConsecutiveDelta2 q,O1 ; :: thesis: S₁[ succ O1]
hence S₁[ succ O1] by A2, XBOOLE_1:1; :: thesis: verum

end;

A3: for O1 being Ordinal st O1 <> {} & O1 is limit_ordinal & ( for O2 being Ordinal st O2 in O1 holds
S₁[O2] ) holds
S₁[O1]

proof

deffunc H₁( Ordinal) -> BiFunction of (ConsecutiveSet2 A,$1),L = ConsecutiveDelta2 q,$1;
let O2 be Ordinal; :: thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O2 being Ordinal st O2 in O2 holds
S₁[O2] ) implies S₁[O2] )
assume that
A4: O2 <> {} and
A5: O2 is limit_ordinal and
for O1 being Ordinal st O1 in O2 holds
d c= ConsecutiveDelta2 q,O1 ; :: thesis: S₁[O2]
A6: {} in O2 by A4, ORDINAL3:10;
consider Ls being T-Sequence such that
A7: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H₁(O1) ) ) from ORDINAL2:sch 2();
Ls . {} = ConsecutiveDelta2 q,{} by A4, A7, ORDINAL3:10
.= d by Th19 ;
then A8: d in rng Ls by A7, A6, FUNCT_1:def 5;
ConsecutiveDelta2 q,O2 = union (rng Ls) by A4, A5, A7, Th21;
hence S₁[O2] by A8, ZFMISC_1:92; :: thesis: verum

end;

A9: S₁[ {} ] by Th19;
for O being Ordinal holds S₁[O] from ORDINAL2:sch 1(A9, A1, A3);
hence d c= ConsecutiveDelta2 q,O ; :: thesis: verum