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 d c= ConsecutiveDelta 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 d c= ConsecutiveDelta q,O
let O be Ordinal; :: thesis: for d being BiFunction of A,L
for q being QuadrSeq of d holds d c= ConsecutiveDelta q,O
let d be BiFunction of A,L; :: thesis: for q being QuadrSeq of d holds d c= ConsecutiveDelta q,O
let q be QuadrSeq of d; :: thesis: d c= ConsecutiveDelta q,O
defpred S₁[ Ordinal] means d c= ConsecutiveDelta q,$1;
A1: for O2 being Ordinal st O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S₁[O1] ) holds
S₁[O2]

proof

deffunc H₁( Ordinal) -> BiFunction of (ConsecutiveSet A,$1),L = ConsecutiveDelta q,$1;
let O2 be Ordinal; :: thesis: ( O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S₁[O1] ) implies S₁[O2] )
assume that
A2: O2 <> {} and
A3: O2 is limit_ordinal and
for O1 being Ordinal st O1 in O2 holds
d c= ConsecutiveDelta q,O1 ; :: thesis: S₁[O2]
A4: {} in O2 by A2, ORDINAL3:10;
consider Ls being T-Sequence such that
A5: ( dom Ls = O2 & ( for O1 being Ordinal st O1 in O2 holds
Ls . O1 = H₁(O1) ) ) from ORDINAL2:sch 2();
Ls . {} = ConsecutiveDelta q,{} by A2, A5, ORDINAL3:10
.= d by Th29 ;
then A6: d in rng Ls by A5, A4, FUNCT_1:def 5;
ConsecutiveDelta q,O2 = union (rng Ls) by A2, A3, A5, Th31;
hence S₁[O2] by A6, ZFMISC_1:92; :: thesis: verum

end;

A7: for O1 being Ordinal st S₁[O1] holds
S₁[ succ O1]

proof

let O1 be Ordinal; :: thesis: ( S₁[O1] implies S₁[ succ O1] )
ConsecutiveDelta q,(succ O1) = new_bi_fun (BiFun (ConsecutiveDelta q,O1),(ConsecutiveSet A,O1),L),(Quadr q,O1) by Th30
.= new_bi_fun (ConsecutiveDelta q,O1),(Quadr q,O1) by Def16 ;
then A8: ConsecutiveDelta q,O1 c= ConsecutiveDelta q,(succ O1) by Th22;
assume d c= ConsecutiveDelta q,O1 ; :: thesis: S₁[ succ O1]
hence S₁[ succ O1] by A8, XBOOLE_1:1; :: thesis: verum

end;

A9: S₁[ {} ] by Th29;
for O being Ordinal holds S₁[O] from ORDINAL2:sch 1(A9, A7, A1);
hence d c= ConsecutiveDelta q,O ; :: thesis: verum