:: YELLOW_1 semantic presentation
:: deftheorem Def1 defines InclPoset YELLOW_1:def 1 :
theorem Th1: :: YELLOW_1:1
:: deftheorem Def2 defines BoolePoset YELLOW_1:def 2 :
theorem Th2: :: YELLOW_1:2
theorem Th3: :: YELLOW_1:3
theorem Th4: :: YELLOW_1:4
theorem Th5: :: YELLOW_1:5
theorem Th6: :: YELLOW_1:6
theorem Th7: :: YELLOW_1:7
theorem Th8: :: YELLOW_1:8
theorem Th9: :: YELLOW_1:9
theorem Th10: :: YELLOW_1:10
theorem Th11: :: YELLOW_1:11
theorem Th12: :: YELLOW_1:12
theorem Th13: :: YELLOW_1:13
theorem Th14: :: YELLOW_1:14
theorem Th15: :: YELLOW_1:15
theorem Th16: :: YELLOW_1:16
Lemma8:
for b1 being set holds the carrier of (BoolePoset b1) = bool b1
by LATTICE3:def 1;
Lemma9:
for b1 being set
for b2, b3 being Element of (BoolePoset b1) holds
( b2 /\ b3 in bool b1 & b2 \/ b3 in bool b1 )
theorem Th17: :: YELLOW_1:17
theorem Th18: :: YELLOW_1:18
theorem Th19: :: YELLOW_1:19
theorem Th20: :: YELLOW_1:20
theorem Th21: :: YELLOW_1:21
theorem Th22: :: YELLOW_1:22
theorem Th23: :: YELLOW_1:23
Lemma10:
for b1 being non empty TopSpace holds InclPoset the topology of b1 is lower-bounded
theorem Th24: :: YELLOW_1:24
Lemma11:
for b1 being non empty TopSpace holds InclPoset the topology of b1 is complete
Lemma12:
for b1 being non empty TopSpace holds
not InclPoset the topology of b1 is trivial
theorem Th25: :: YELLOW_1:25
:: deftheorem Def3 defines RelStr-yielding YELLOW_1:def 3 :
definition
let c
1 be
set ;
let c
2 be
RelStr-yielding ManySortedSet of c
1;
func product c
2 -> strict RelStr means :
Def4:
:: YELLOW_1:def 4
( the
carrier of a
3 = product (Carrier a2) & ( for b
1, b
2 being
Element of a
3 holds
( b
1 in product (Carrier a2) implies ( b
1 <= b
2 iff ex b
3, b
4 being
Function st
( b
3 = b
1 & b
4 = b
2 & ( for b
5 being
set holds
not ( b
5 in a
1 & ( for b
6 being
RelStr for b
7, b
8 being
Element of b
6 holds
not ( b
6 = a
2 . b
5 & b
7 = b
3 . b
5 & b
8 = b
4 . b
5 & b
7 <= b
8 ) ) ) ) ) ) ) ) );
existence
ex b1 being strict RelStr st
( the carrier of b1 = product (Carrier c2) & ( for b2, b3 being Element of b1 holds
( b2 in product (Carrier c2) implies ( b2 <= b3 iff ex b4, b5 being Function st
( b4 = b2 & b5 = b3 & ( for b6 being set holds
not ( b6 in c1 & ( for b7 being RelStr
for b8, b9 being Element of b7 holds
not ( b7 = c2 . b6 & b8 = b4 . b6 & b9 = b5 . b6 & b8 <= b9 ) ) ) ) ) ) ) ) )
uniqueness
for b1, b2 being strict RelStr holds
( the carrier of b1 = product (Carrier c2) & ( for b3, b4 being Element of b1 holds
( b3 in product (Carrier c2) implies ( b3 <= b4 iff ex b5, b6 being Function st
( b5 = b3 & b6 = b4 & ( for b7 being set holds
not ( b7 in c1 & ( for b8 being RelStr
for b9, b10 being Element of b8 holds
not ( b8 = c2 . b7 & b9 = b5 . b7 & b10 = b6 . b7 & b9 <= b10 ) ) ) ) ) ) ) ) & the carrier of b2 = product (Carrier c2) & ( for b3, b4 being Element of b2 holds
( b3 in product (Carrier c2) implies ( b3 <= b4 iff ex b5, b6 being Function st
( b5 = b3 & b6 = b4 & ( for b7 being set holds
not ( b7 in c1 & ( for b8 being RelStr
for b9, b10 being Element of b8 holds
not ( b8 = c2 . b7 & b9 = b5 . b7 & b10 = b6 . b7 & b9 <= b10 ) ) ) ) ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines product YELLOW_1:def 4 :
:: deftheorem Def5 defines |^ YELLOW_1:def 5 :
theorem Th26: :: YELLOW_1:26
theorem Th27: :: YELLOW_1:27
theorem Th28: :: YELLOW_1:28
Lemma18:
for b1 being set
for b2 being non empty RelStr
for b3 being Element of (b2 |^ b1) holds
( b3 in the carrier of (product (b1 --> b2)) & b3 is Function of b1,the carrier of b2 )
definition
let c
1, c
2 be
RelStr ;
func MonMaps c
1,c
2 -> strict full SubRelStr of a
2 |^ the
carrier of a
1 means :: YELLOW_1:def 6
for b
1 being
Function of a
1,a
2 holds
( b
1 in the
carrier of a
3 iff ( b
1 in Funcs the
carrier of a
1,the
carrier of a
2 & b
1 is
monotone ) );
existence
ex b1 being strict full SubRelStr of c2 |^ the carrier of c1 st
for b2 being Function of c1,c2 holds
( b2 in the carrier of b1 iff ( b2 in Funcs the carrier of c1,the carrier of c2 & b2 is monotone ) )
uniqueness
for b1, b2 being strict full SubRelStr of c2 |^ the carrier of c1 holds
( ( for b3 being Function of c1,c2 holds
( b3 in the carrier of b1 iff ( b3 in Funcs the carrier of c1,the carrier of c2 & b3 is monotone ) ) ) & ( for b3 being Function of c1,c2 holds
( b3 in the carrier of b2 iff ( b3 in Funcs the carrier of c1,the carrier of c2 & b3 is monotone ) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines MonMaps YELLOW_1:def 6 :