:: TOPREAL7 semantic presentation
theorem Th1: :: TOPREAL7:1
for b
1, b
2 being
Real holds
(
max b
1,b
2 <= b
1 implies
max b
1,b
2 = b
1 )
theorem Th2: :: TOPREAL7:2
theorem Th3: :: TOPREAL7:3
for b
1, b
2, b
3, b
4, b
5, b
6 being
Real holds
( b
1 <= b
2 + b
3 & b
4 <= b
5 + b
6 implies
max b
1,b
4 <= (max b2,b5) + (max b3,b6) )
theorem Th4: :: TOPREAL7:4
theorem Th5: :: TOPREAL7:5
theorem Th6: :: TOPREAL7:6
theorem Th7: :: TOPREAL7:7
theorem Th8: :: TOPREAL7:8
theorem Th9: :: TOPREAL7:9
theorem Th10: :: TOPREAL7:10
theorem Th11: :: TOPREAL7:11
theorem Th12: :: TOPREAL7:12
theorem Th13: :: TOPREAL7:13
theorem Th14: :: TOPREAL7:14
theorem Th15: :: TOPREAL7:15
theorem Th16: :: TOPREAL7:16
theorem Th17: :: TOPREAL7:17
theorem Th18: :: TOPREAL7:18
theorem Th19: :: TOPREAL7:19
definition
let c
1, c
2 be non
empty MetrStruct ;
func max-Prod2 c
1,c
2 -> strict MetrStruct means :
Def1:
:: TOPREAL7:def 1
( the
carrier of a
3 = [:the carrier of a1,the carrier of a2:] & ( for b
1, b
2 being
Point of a
3 holds
ex b
3, b
4 being
Point of a
1ex b
5, b
6 being
Point of a
2 st
( b
1 = [b3,b5] & b
2 = [b4,b6] & the
distance of a
3 . b
1,b
2 = max (the distance of a1 . b3,b4),
(the distance of a2 . b5,b6) ) ) );
existence
ex b1 being strict MetrStruct st
( the carrier of b1 = [:the carrier of c1,the carrier of c2:] & ( for b2, b3 being Point of b1 holds
ex b4, b5 being Point of c1ex b6, b7 being Point of c2 st
( b2 = [b4,b6] & b3 = [b5,b7] & the distance of b1 . b2,b3 = max (the distance of c1 . b4,b5),(the distance of c2 . b6,b7) ) ) )
uniqueness
for b1, b2 being strict MetrStruct holds
( the carrier of b1 = [:the carrier of c1,the carrier of c2:] & ( for b3, b4 being Point of b1 holds
ex b5, b6 being Point of c1ex b7, b8 being Point of c2 st
( b3 = [b5,b7] & b4 = [b6,b8] & the distance of b1 . b3,b4 = max (the distance of c1 . b5,b6),(the distance of c2 . b7,b8) ) ) & the carrier of b2 = [:the carrier of c1,the carrier of c2:] & ( for b3, b4 being Point of b2 holds
ex b5, b6 being Point of c1ex b7, b8 being Point of c2 st
( b3 = [b5,b7] & b4 = [b6,b8] & the distance of b2 . b3,b4 = max (the distance of c1 . b5,b6),(the distance of c2 . b7,b8) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines max-Prod2 TOPREAL7:def 1 :
for b
1, b
2 being non
empty MetrStruct for b
3 being
strict MetrStruct holds
( b
3 = max-Prod2 b
1,b
2 iff ( the
carrier of b
3 = [:the carrier of b1,the carrier of b2:] & ( for b
4, b
5 being
Point of b
3 holds
ex b
6, b
7 being
Point of b
1ex b
8, b
9 being
Point of b
2 st
( b
4 = [b6,b8] & b
5 = [b7,b9] & the
distance of b
3 . b
4,b
5 = max (the distance of b1 . b6,b7),
(the distance of b2 . b8,b9) ) ) ) );
theorem Th20: :: TOPREAL7:20
theorem Th21: :: TOPREAL7:21
theorem Th22: :: TOPREAL7:22
Lemma17:
for b1, b2 being non empty MetrSpace holds max-Prod2 b1,b2 is discerning
theorem Th23: :: TOPREAL7:23
theorem Th24: :: TOPREAL7:24
theorem Th25: :: TOPREAL7:25
theorem Th26: :: TOPREAL7:26
theorem Th27: :: TOPREAL7:27