:: TOPRNS_1 semantic presentation
theorem Th1: :: TOPRNS_1:1
canceled;
theorem Th2: :: TOPRNS_1:2
:: deftheorem Def1 defines non-zero TOPRNS_1:def 1 :
theorem Th3: :: TOPRNS_1:3
theorem Th4: :: TOPRNS_1:4
theorem Th5: :: TOPRNS_1:5
for b
1 being
Natfor b
2, b
3 being
Real_Sequence of b
1 holds
( ( for b
4 being
set holds
( b
4 in NAT implies b
2 . b
4 = b
3 . b
4 ) ) implies b
2 = b
3 )
theorem Th6: :: TOPRNS_1:6
for b
1 being
Natfor b
2, b
3 being
Real_Sequence of b
1 holds
( ( for b
4 being
Nat holds b
2 . b
4 = b
3 . b
4 ) implies b
2 = b
3 )
:: deftheorem Def2 defines + TOPRNS_1:def 2 :
for b
1 being
Natfor b
2, b
3, b
4 being
Real_Sequence of b
1 holds
( b
4 = b
2 + b
3 iff for b
5 being
Nat holds b
4 . b
5 = (b2 . b5) + (b3 . b5) );
:: deftheorem Def3 defines * TOPRNS_1:def 3 :
for b
1 being
Realfor b
2 being
Natfor b
3, b
4 being
Real_Sequence of b
2 holds
( b
4 = b
1 * b
3 iff for b
5 being
Nat holds b
4 . b
5 = b
1 * (b3 . b5) );
:: deftheorem Def4 defines - TOPRNS_1:def 4 :
:: deftheorem Def5 defines - TOPRNS_1:def 5 :
:: deftheorem Def6 defines |. TOPRNS_1:def 6 :
:: deftheorem Def7 defines |. TOPRNS_1:def 7 :
theorem Th7: :: TOPRNS_1:7
canceled;
theorem Th8: :: TOPRNS_1:8
theorem Th9: :: TOPRNS_1:9
theorem Th10: :: TOPRNS_1:10
theorem Th11: :: TOPRNS_1:11
theorem Th12: :: TOPRNS_1:12
theorem Th13: :: TOPRNS_1:13
theorem Th14: :: TOPRNS_1:14
theorem Th15: :: TOPRNS_1:15
theorem Th16: :: TOPRNS_1:16
theorem Th17: :: TOPRNS_1:17
theorem Th18: :: TOPRNS_1:18
theorem Th19: :: TOPRNS_1:19
theorem Th20: :: TOPRNS_1:20
theorem Th21: :: TOPRNS_1:21
theorem Th22: :: TOPRNS_1:22
theorem Th23: :: TOPRNS_1:23
theorem Th24: :: TOPRNS_1:24
theorem Th25: :: TOPRNS_1:25
theorem Th26: :: TOPRNS_1:26
theorem Th27: :: TOPRNS_1:27
theorem Th28: :: TOPRNS_1:28
Lemma27:
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1) holds
( |.(b2 - b3).| = 0 implies b2 = b3 )
Lemma28:
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1) holds
( b2 = b3 implies |.(b2 - b3).| = 0 )
theorem Th29: :: TOPRNS_1:29
theorem Th30: :: TOPRNS_1:30
theorem Th31: :: TOPRNS_1:31
theorem Th32: :: TOPRNS_1:32
theorem Th33: :: TOPRNS_1:33
theorem Th34: :: TOPRNS_1:34
theorem Th35: :: TOPRNS_1:35
theorem Th36: :: TOPRNS_1:36
theorem Th37: :: TOPRNS_1:37
canceled;
theorem Th38: :: TOPRNS_1:38
:: deftheorem Def8 defines bounded TOPRNS_1:def 8 :
theorem Th39: :: TOPRNS_1:39
:: deftheorem Def9 defines convergent TOPRNS_1:def 9 :
:: deftheorem Def10 defines lim TOPRNS_1:def 10 :
theorem Th40: :: TOPRNS_1:40
canceled;
theorem Th41: :: TOPRNS_1:41
theorem Th42: :: TOPRNS_1:42
theorem Th43: :: TOPRNS_1:43
theorem Th44: :: TOPRNS_1:44
theorem Th45: :: TOPRNS_1:45
theorem Th46: :: TOPRNS_1:46
theorem Th47: :: TOPRNS_1:47
theorem Th48: :: TOPRNS_1:48
theorem Th49: :: TOPRNS_1:49
canceled;
theorem Th50: :: TOPRNS_1:50
theorem Th51: :: TOPRNS_1:51