let A be non empty set ; :: thesis: for L being lower-bounded LATTICE
for d being BiFunction of A,L
for O1, O2 being Ordinal
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2
let L be lower-bounded LATTICE; :: thesis: for d being BiFunction of A,L
for O1, O2 being Ordinal
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2
let d be BiFunction of A,L; :: thesis: for O1, O2 being Ordinal
for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2
let O1, O2 be Ordinal; :: thesis: for q being QuadrSeq of d st O1 c= O2 holds
ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2
let q be QuadrSeq of d; :: thesis: ( O1 c= O2 implies ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2 )
defpred S₁[ Ordinal] means ( O1 c= $1 implies ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,$1 );
A1: 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
A2: ( O2 <> {} & O2 is limit_ordinal ) and
for O3 being Ordinal st O3 in O2 & O1 c= O3 holds
ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O3 ; :: thesis: S₁[O2]
consider L being T-Sequence such that
A3: ( dom L = O2 & ( for O3 being Ordinal st O3 in O2 holds
L . O3 = H₁(O3) ) ) from ORDINAL2:sch 2();
A4: ConsecutiveDelta2 q,O2 = union (rng L) by A2, A3, Th21;
assume A5: O1 c= O2 ; :: thesis: ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2

per cases ( O1 = O2 or O1 <> O2 ) ;

suppose O1 = O2 ; :: thesis: ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2

hence ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2 ; :: thesis: verum

end;

suppose O1 <> O2 ; :: thesis: ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2

then A6: O1 c< O2 by A5, XBOOLE_0:def 8;
then O1 in O2 by ORDINAL1:21;
then A7: L . O1 in rng L by A3, FUNCT_1:def 5;
L . O1 = ConsecutiveDelta2 q,O1 by A3, A6, ORDINAL1:21;
hence ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2 by A4, A7, ZFMISC_1:92; :: thesis: verum

end;

end;

end;

A8: for O1 being Ordinal st S₁[O1] holds
S₁[ succ O1]

proof

let O2 be Ordinal; :: thesis: ( S₁[O2] implies S₁[ succ O2] )
assume A9: ( O1 c= O2 implies ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2 ) ; :: thesis: S₁[ succ O2]
assume A10: O1 c= succ O2 ; :: thesis: ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,(succ O2)

per cases ( O1 = succ O2 or O1 <> succ O2 ) ;

suppose O1 = succ O2 ; :: thesis: ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,(succ O2)

hence ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,(succ O2) ; :: thesis: verum

end;

suppose A11: O1 <> succ O2 ; :: thesis: ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,(succ O2)

ConsecutiveDelta2 q,(succ O2) = new_bi_fun2 (BiFun (ConsecutiveDelta2 q,O2),(ConsecutiveSet2 A,O2),L),(Quadr2 q,O2) by Th20
.= new_bi_fun2 (ConsecutiveDelta2 q,O2),(Quadr2 q,O2) by LATTICE5:def 16 ;
then A12: ConsecutiveDelta2 q,O2 c= ConsecutiveDelta2 q,(succ O2) by Th14;
O1 c< succ O2 by A10, A11, XBOOLE_0:def 8;
then O1 in succ O2 by ORDINAL1:21;
hence ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,(succ O2) by A9, A12, ORDINAL1:34, XBOOLE_1:1; :: thesis: verum

end;

end;

end;

A13: S₁[ {} ] ;
for O being Ordinal holds S₁[O] from ORDINAL2:sch 1(A13, A8, A1);
hence ( O1 c= O2 implies ConsecutiveDelta2 q,O1 c= ConsecutiveDelta2 q,O2 ) ; :: thesis: verum