let A be non empty set ; 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; 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; for q being QuadrSeq of d
for O being Ordinal holds d c= ConsecutiveDelta2 (q,O)
let q be QuadrSeq of d; for O being Ordinal holds d c= ConsecutiveDelta2 (q,O)
let O be Ordinal; 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;
( 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 Th19
.=
new_bi_fun2 (
(ConsecutiveDelta2 (q,O1)),
(Quadr2 (q,O1)))
by LATTICE5:def 15
;
then A2:
ConsecutiveDelta2 (
q,
O1)
c= ConsecutiveDelta2 (
q,
(succ O1))
by Th13;
assume
d c= ConsecutiveDelta2 (
q,
O1)
;
S1[ succ O1]
hence
S1[
succ O1]
by A2, XBOOLE_1:1;
verum
end;
A3:
for O1 being Ordinal st O1 <> 0 & 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;
( O2 <> 0 & O2 is limit_ordinal & ( for O2 being Ordinal st O2 in O2 holds
S1[O2] ) implies S1[O2] )
assume that A4:
O2 <> 0
and A5:
O2 is
limit_ordinal
and
for
O1 being
Ordinal st
O1 in O2 holds
d c= ConsecutiveDelta2 (
q,
O1)
;
S1[O2]
A6:
{} in O2
by A4, ORDINAL3:8;
consider Ls being
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, ORDINAL3:8
.=
d
by Th18
;
then A8:
d in rng Ls
by A7, A6, FUNCT_1:def 3;
ConsecutiveDelta2 (
q,
O2)
= union (rng Ls)
by A4, A5, A7, Th20;
hence
S1[
O2]
by A8, ZFMISC_1:74;
verum
end;
A9:
S1[ 0 ]
by Th18;
for O being Ordinal holds S1[O]
from ORDINAL2:sch 1(A9, A1, A3);
hence
d c= ConsecutiveDelta2 (q,O)
; verum