let A be non empty set ; for L being lower-bounded LATTICE
for O being Ordinal
for d being BiFunction of A,L st d is zeroed holds
for q being QuadrSeq of d holds ConsecutiveDelta q,O is zeroed
let L be lower-bounded LATTICE; for O being Ordinal
for d being BiFunction of A,L st d is zeroed holds
for q being QuadrSeq of d holds ConsecutiveDelta q,O is zeroed
let O be Ordinal; for d being BiFunction of A,L st d is zeroed holds
for q being QuadrSeq of d holds ConsecutiveDelta q,O is zeroed
let d be BiFunction of A,L; ( d is zeroed implies for q being QuadrSeq of d holds ConsecutiveDelta q,O is zeroed )
assume A1:
d is zeroed
; for q being QuadrSeq of d holds ConsecutiveDelta q,O is zeroed
let q be QuadrSeq of d; ConsecutiveDelta q,O is zeroed
defpred S1[ Ordinal] means ConsecutiveDelta q,$1 is zeroed ;
A2:
for O1 being Ordinal st S1[O1] holds
S1[ succ O1]
proof
let O1 be
Ordinal;
( S1[O1] implies S1[ succ O1] )
assume
ConsecutiveDelta q,
O1 is
zeroed
;
S1[ succ O1]
then A3:
new_bi_fun (ConsecutiveDelta q,O1),
(Quadr q,O1) is
zeroed
by Th18;
let z be
Element of
ConsecutiveSet A,
(succ O1);
LATTICE5:def 7 (ConsecutiveDelta q,(succ O1)) . z,z = Bottom L
reconsider z9 =
z as
Element of
new_set (ConsecutiveSet A,O1) by Th25;
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
;
hence (ConsecutiveDelta q,(succ O1)) . z,
z =
(new_bi_fun (ConsecutiveDelta q,O1),(Quadr q,O1)) . z9,
z9
.=
Bottom L
by A3, Def7
;
verum
end;
A4:
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;
( O2 <> {} & O2 is limit_ordinal & ( for O1 being Ordinal st O1 in O2 holds
S1[O1] ) implies S1[O2] )
assume that A5:
(
O2 <> {} &
O2 is
limit_ordinal )
and A6:
for
O1 being
Ordinal st
O1 in O2 holds
ConsecutiveDelta q,
O1 is
zeroed
;
S1[O2]
set CS =
ConsecutiveSet A,
O2;
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();
ConsecutiveDelta q,
O2 = union (rng Ls)
by A5, A7, Th31;
then reconsider f =
union (rng Ls) as
BiFunction of
(ConsecutiveSet A,O2),
L ;
deffunc H2(
Ordinal)
-> set =
ConsecutiveSet A,$1;
consider Ts being
T-Sequence such that A8:
(
dom Ts = O2 & ( for
O1 being
Ordinal st
O1 in O2 holds
Ts . O1 = H2(
O1) ) )
from ORDINAL2:sch 2();
A9:
ConsecutiveSet A,
O2 = union (rng Ts)
by A5, A8, Th26;
f is
zeroed
proof
let x be
Element of
ConsecutiveSet A,
O2;
LATTICE5:def 7 f . x,x = Bottom L
consider y being
set such that A10:
x in y
and A11:
y in rng Ts
by A9, TARSKI:def 4;
consider o being
set such that A12:
o in dom Ts
and A13:
y = Ts . o
by A11, FUNCT_1:def 5;
reconsider o =
o as
Ordinal by A12;
A14:
Ls . o = ConsecutiveDelta q,
o
by A7, A8, A12;
then reconsider h =
Ls . o as
BiFunction of
(ConsecutiveSet A,o),
L ;
reconsider x9 =
x as
Element of
ConsecutiveSet A,
o by A8, A10, A12, A13;
A15:
dom h = [:(ConsecutiveSet A,o),(ConsecutiveSet A,o):]
by FUNCT_2:def 1;
A16:
h is
zeroed
proof
let z be
Element of
ConsecutiveSet A,
o;
LATTICE5:def 7 h . z,z = Bottom L
A17:
ConsecutiveDelta q,
o is
zeroed
by A6, A8, A12;
thus h . z,
z =
(ConsecutiveDelta q,o) . z,
z
by A7, A8, A12
.=
Bottom L
by A17, Def7
;
verum
end;
ConsecutiveDelta q,
o in rng Ls
by A7, A8, A12, A14, FUNCT_1:def 5;
then A18:
h c= f
by A14, ZFMISC_1:92;
x in ConsecutiveSet A,
o
by A8, A10, A12, A13;
then
[x,x] in dom h
by A15, ZFMISC_1:106;
hence f . x,
x =
h . x9,
x9
by A18, GRFUNC_1:8
.=
Bottom L
by A16, Def7
;
verum
end;
hence
S1[
O2]
by A5, A7, Th31;
verum
end;
A19:
S1[ {} ]
for O being Ordinal holds S1[O]
from ORDINAL2:sch 1(A19, A2, A4);
hence
ConsecutiveDelta q,O is zeroed
; verum