:: CLOSURE1 semantic presentation
theorem Th1: :: CLOSURE1:1
theorem Th2: :: CLOSURE1:2
E49:
now
let I be
set ;
let A be
V5 ManySortedSet of
I,
B be
V5 ManySortedSet of
I;
let F be
ManySortedFunction of
A,
B;
let X be
Element of
A;
thus
F .. X is
Element of
B
proof
let i be
set ;
:: according to PBOOLE:def 17
assume E34:
i in I
;
E36:
dom F = I
by PBOOLE:def 3;
reconsider g =
F . i as
Function ;
E37:
g is
Function of
A . i,
B . i
by , PBOOLE:def 18;
E38:
A . i <> {}
by , PBOOLE:def 16;
E39:
B . i <> {}
by , PBOOLE:def 16;
X . i is
Element of
A . i
by , PBOOLE:def 17;
then
g . (X . i) in B . i
by , , , FUNCT_2:7;
hence
(F .. X) . i is
Element of
B . i
by , , PRALG_1:def 17;
end;
end;
theorem Th3: :: CLOSURE1:3
theorem Th4: :: CLOSURE1:4
theorem Th5: :: CLOSURE1:5
theorem Th6: :: CLOSURE1:6
theorem Th7: :: CLOSURE1:7
:: deftheorem Def1 CLOSURE1:def 1 :
canceled;
:: deftheorem Def2 defines reflexive CLOSURE1:def 2 :
:: deftheorem Def3 defines monotonic CLOSURE1:def 3 :
:: deftheorem Def4 defines idempotent CLOSURE1:def 4 :
:: deftheorem Def5 defines topological CLOSURE1:def 5 :
theorem Th8: :: CLOSURE1:8
theorem Th9: :: CLOSURE1:9
theorem Th10: :: CLOSURE1:10
theorem Th11: :: CLOSURE1:11
theorem Th12: :: CLOSURE1:12
theorem Th13: :: CLOSURE1:13
theorem Th14: :: CLOSURE1:14
theorem Th15: :: CLOSURE1:15
theorem Th16: :: CLOSURE1:16
theorem Th17: :: CLOSURE1:17
theorem Th18: :: CLOSURE1:18
theorem Th19: :: CLOSURE1:19
theorem Th20: :: CLOSURE1:20
theorem Th21: :: CLOSURE1:21
theorem Th22: :: CLOSURE1:22
theorem Th23: :: CLOSURE1:23
theorem Th24: :: CLOSURE1:24
theorem Th25: :: CLOSURE1:25
theorem Th26: :: CLOSURE1:26
theorem Th27: :: CLOSURE1:27
:: deftheorem Def6 defines additive CLOSURE1:def 6 :
:: deftheorem Def7 defines absolutely-additive CLOSURE1:def 7 :
:: deftheorem Def8 defines multiplicative CLOSURE1:def 8 :
:: deftheorem Def9 defines absolutely-multiplicative CLOSURE1:def 9 :
:: deftheorem Def10 defines properly-upper-bound CLOSURE1:def 10 :
:: deftheorem Def11 defines properly-lower-bound CLOSURE1:def 11 :
:: deftheorem Def12 defines MSFull CLOSURE1:def 12 :
:: deftheorem Def13 defines MSFixPoints CLOSURE1:def 13 :
theorem Th28: :: CLOSURE1:28
theorem Th29: :: CLOSURE1:29
theorem Th30: :: CLOSURE1:30
theorem Th31: :: CLOSURE1:31
theorem Th32: :: CLOSURE1:32
theorem Th33: :: CLOSURE1:33
theorem Th34: :: CLOSURE1:34
theorem Th35: :: CLOSURE1:35
theorem Th36: :: CLOSURE1:36
theorem Th37: :: CLOSURE1:37
theorem Th38: :: CLOSURE1:38
:: deftheorem Def14 defines ClOp->ClSys CLOSURE1:def 14 :
:: deftheorem Def15 defines ClSys->ClOp CLOSURE1:def 15 :
theorem Th39: :: CLOSURE1:39
theorem Th40: :: CLOSURE1:40