:: WAYBEL20 semantic presentation
theorem Th1: :: WAYBEL20:1
theorem Th2: :: WAYBEL20:2
definition
let c
1, c
2, c
3, c
4 be
RelStr ;
let c
5 be
Function of c
1,c
3;
let c
6 be
Function of c
2,c
4;
redefine func [: as
[:c5,c6:] -> Function of
[:a1,a2:],
[:a3,a4:];
coherence
[:c5,c6:] is Function of [:c1,c2:],[:c3,c4:]
end;
theorem Th3: :: WAYBEL20:3
theorem Th4: :: WAYBEL20:4
theorem Th5: :: WAYBEL20:5
theorem Th6: :: WAYBEL20:6
theorem Th7: :: WAYBEL20:7
theorem Th8: :: WAYBEL20:8
theorem Th9: :: WAYBEL20:9
theorem Th10: :: WAYBEL20:10
theorem Th11: :: WAYBEL20:11
theorem Th12: :: WAYBEL20:12
theorem Th13: :: WAYBEL20:13
theorem Th14: :: WAYBEL20:14
theorem Th15: :: WAYBEL20:15
theorem Th16: :: WAYBEL20:16
theorem Th17: :: WAYBEL20:17
theorem Th18: :: WAYBEL20:18
theorem Th19: :: WAYBEL20:19
theorem Th20: :: WAYBEL20:20
theorem Th21: :: WAYBEL20:21
theorem Th22: :: WAYBEL20:22
theorem Th23: :: WAYBEL20:23
canceled;
theorem Th24: :: WAYBEL20:24
theorem Th25: :: WAYBEL20:25
theorem Th26: :: WAYBEL20:26
theorem Th27: :: WAYBEL20:27
theorem Th28: :: WAYBEL20:28
theorem Th29: :: WAYBEL20:29
theorem Th30: :: WAYBEL20:30
theorem Th31: :: WAYBEL20:31
theorem Th32: :: WAYBEL20:32
theorem Th33: :: WAYBEL20:33
theorem Th34: :: WAYBEL20:34
theorem Th35: :: WAYBEL20:35
:: deftheorem Def1 defines EqRel WAYBEL20:def 1 :
:: deftheorem Def2 defines CLCongruence WAYBEL20:def 2 :
theorem Th36: :: WAYBEL20:36
:: deftheorem Def3 defines kernel_op WAYBEL20:def 3 :
theorem Th37: :: WAYBEL20:37
theorem Th38: :: WAYBEL20:38
for b
1 being
complete continuous LATTICEfor b
2 being
Subset of
[:b1,b1:]for b
3 being
kernel Function of b
1,b
1 holds
not ( b
3 is
directed-sups-preserving & b
2 = [:b3,b3:] " (id the carrier of b1) & ( for b
4 being
strict complete continuous LATTICE holds
not ( the
carrier of b
4 = Class (EqRel b2) & the
InternalRel of b
4 = { [(Class (EqRel b2),b5),(Class (EqRel b2),b6)] where B is Element of b1, B is Element of b1 : b3 . b5 <= b3 . b6 } & ( for b
5 being
Function of b
1,b
4 holds
( ( for b
6 being
Element of b
1 holds b
5 . b
6 = Class (EqRel b2),b
6 ) implies b
5 is
CLHomomorphism of b
1,b
4 ) ) ) ) )
theorem Th39: :: WAYBEL20:39
:: deftheorem Def4 defines kernel_congruence WAYBEL20:def 4 :
theorem Th40: :: WAYBEL20:40
theorem Th41: :: WAYBEL20:41
definition
let c
1 be
complete continuous LATTICE;
let c
2 be non
empty Subset of
[:c1,c1:];
assume E32:
c
2 is
CLCongruence
;
func c
1 ./. c
2 -> strict complete continuous LATTICE means :
Def5:
:: WAYBEL20:def 5
( the
carrier of a
3 = Class (EqRel a2) & ( for b
1, b
2 being
Element of a
3 holds
( b
1 <= b
2 iff
"/\" b
1,a
1 <= "/\" b
2,a
1 ) ) );
existence
ex b1 being strict complete continuous LATTICE st
( the carrier of b1 = Class (EqRel c2) & ( for b2, b3 being Element of b1 holds
( b2 <= b3 iff "/\" b2,c1 <= "/\" b3,c1 ) ) )
uniqueness
for b1, b2 being strict complete continuous LATTICE holds
( the carrier of b1 = Class (EqRel c2) & ( for b3, b4 being Element of b1 holds
( b3 <= b4 iff "/\" b3,c1 <= "/\" b4,c1 ) ) & the carrier of b2 = Class (EqRel c2) & ( for b3, b4 being Element of b2 holds
( b3 <= b4 iff "/\" b3,c1 <= "/\" b4,c1 ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines ./. WAYBEL20:def 5 :
theorem Th42: :: WAYBEL20:42
theorem Th43: :: WAYBEL20:43
theorem Th44: :: WAYBEL20:44
theorem Th45: :: WAYBEL20:45