:: PCOMPS_1 semantic presentation
theorem Th1: :: PCOMPS_1:1
theorem Th2: :: PCOMPS_1:2
theorem Th3: :: PCOMPS_1:3
canceled;
theorem Th4: :: PCOMPS_1:4
canceled;
theorem Th5: :: PCOMPS_1:5
:: deftheorem Def1 PCOMPS_1:def 1 :
canceled;
theorem Th6: :: PCOMPS_1:6
canceled;
theorem Th7: :: PCOMPS_1:7
theorem Th8: :: PCOMPS_1:8
theorem Th9: :: PCOMPS_1:9
theorem Th10: :: PCOMPS_1:10
:: deftheorem Def2 defines locally_finite PCOMPS_1:def 2 :
theorem Th11: :: PCOMPS_1:11
theorem Th12: :: PCOMPS_1:12
theorem Th13: :: PCOMPS_1:13
:: deftheorem Def3 defines clf PCOMPS_1:def 3 :
theorem Th14: :: PCOMPS_1:14
theorem Th15: :: PCOMPS_1:15
theorem Th16: :: PCOMPS_1:16
theorem Th17: :: PCOMPS_1:17
theorem Th18: :: PCOMPS_1:18
theorem Th19: :: PCOMPS_1:19
theorem Th20: :: PCOMPS_1:20
theorem Th21: :: PCOMPS_1:21
theorem Th22: :: PCOMPS_1:22
theorem Th23: :: PCOMPS_1:23
theorem Th24: :: PCOMPS_1:24
:: deftheorem Def4 defines paracompact PCOMPS_1:def 4 :
theorem Th25: :: PCOMPS_1:25
theorem Th26: :: PCOMPS_1:26
theorem Th27: :: PCOMPS_1:27
theorem Th28: :: PCOMPS_1:28
:: deftheorem Def5 defines Family_open_set PCOMPS_1:def 5 :
theorem Th29: :: PCOMPS_1:29
theorem Th30: :: PCOMPS_1:30
theorem Th31: :: PCOMPS_1:31
theorem Th32: :: PCOMPS_1:32
canceled;
theorem Th33: :: PCOMPS_1:33
theorem Th34: :: PCOMPS_1:34
theorem Th35: :: PCOMPS_1:35
theorem Th36: :: PCOMPS_1:36
theorem Th37: :: PCOMPS_1:37
:: deftheorem Def6 defines TopSpaceMetr PCOMPS_1:def 6 :
theorem Th38: :: PCOMPS_1:38
definition
let c
1 be
set ;
let c
2 be
Function of
[:c1,c1:],
REAL ;
pred c
2 is_metric_of c
1 means :
Def7:
:: PCOMPS_1:def 7
for b
1, b
2, b
3 being
Element of a
1 holds
( ( a
2 . b
1,b
2 = 0 implies b
1 = b
2 ) & ( b
1 = b
2 implies a
2 . b
1,b
2 = 0 ) & a
2 . b
1,b
2 = a
2 . b
2,b
1 & a
2 . b
1,b
3 <= (a2 . b1,b2) + (a2 . b2,b3) );
end;
:: deftheorem Def7 defines is_metric_of PCOMPS_1:def 7 :
for b
1 being
set for b
2 being
Function of
[:b1,b1:],
REAL holds
( b
2 is_metric_of b
1 iff for b
3, b
4, b
5 being
Element of b
1 holds
( ( b
2 . b
3,b
4 = 0 implies b
3 = b
4 ) & ( b
3 = b
4 implies b
2 . b
3,b
4 = 0 ) & b
2 . b
3,b
4 = b
2 . b
4,b
3 & b
2 . b
3,b
5 <= (b2 . b3,b4) + (b2 . b4,b5) ) );
theorem Th39: :: PCOMPS_1:39
:: deftheorem Def8 defines SpaceMetr PCOMPS_1:def 8 :
:: deftheorem Def9 defines metrizable PCOMPS_1:def 9 :