:: Real Sequences and Basic Operations on Them
:: by Jaros{\l}aw Kotowicz
::
:: Received July 4, 1989
:: Copyright (c) 1990 Association of Mizar Users
theorem :: SEQ_1:1
canceled;
theorem :: SEQ_1:2
canceled;
theorem Th3: :: SEQ_1:3
theorem Th4: :: SEQ_1:4
theorem :: SEQ_1:5
theorem Th6: :: SEQ_1:6
theorem Th7: :: SEQ_1:7
theorem :: SEQ_1:8
canceled;
theorem :: SEQ_1:9
canceled;
theorem :: SEQ_1:10
theorem Th11: :: SEQ_1:11
theorem Th12: :: SEQ_1:12
theorem Th13: :: SEQ_1:13
theorem :: SEQ_1:14
theorem :: SEQ_1:15
theorem Th16: :: SEQ_1:16
theorem :: SEQ_1:17
canceled;
theorem :: SEQ_1:18
canceled;
theorem :: SEQ_1:19
canceled;
theorem Th20: :: SEQ_1:20
for
seq1,
seq2,
seq3 being
Real_Sequence holds
(seq1 + seq2) + seq3 = seq1 + (seq2 + seq3)
theorem :: SEQ_1:21
canceled;
theorem Th22: :: SEQ_1:22
theorem Th23: :: SEQ_1:23
theorem :: SEQ_1:24
theorem :: SEQ_1:25
theorem Th26: :: SEQ_1:26
theorem Th27: :: SEQ_1:27
theorem Th28: :: SEQ_1:28
theorem :: SEQ_1:29
theorem Th30: :: SEQ_1:30
theorem Th31: :: SEQ_1:31
theorem Th32: :: SEQ_1:32
theorem :: SEQ_1:33
theorem :: SEQ_1:34
for
seq1,
seq2,
seq3 being
Real_Sequence holds
seq1 - (seq2 + seq3) = (seq1 - seq2) - seq3
theorem :: SEQ_1:35
theorem :: SEQ_1:36
theorem :: SEQ_1:37
theorem :: SEQ_1:38
for
seq1,
seq2,
seq3 being
Real_Sequence holds
seq1 - (seq2 - seq3) = (seq1 - seq2) + seq3
theorem :: SEQ_1:39
for
seq1,
seq2,
seq3 being
Real_Sequence holds
seq1 + (seq2 - seq3) = (seq1 + seq2) - seq3
theorem :: SEQ_1:40
theorem Th41: :: SEQ_1:41
theorem :: SEQ_1:42
theorem Th43: :: SEQ_1:43
theorem Th44: :: SEQ_1:44
theorem :: SEQ_1:45
theorem :: SEQ_1:46
theorem :: SEQ_1:47
theorem Th48: :: SEQ_1:48
theorem :: SEQ_1:49
theorem :: SEQ_1:50
theorem Th51: :: SEQ_1:51
theorem Th52: :: SEQ_1:52
theorem :: SEQ_1:53
theorem Th54: :: SEQ_1:54
Lm1:
(- 1) " = - 1
;
theorem :: SEQ_1:55
theorem :: SEQ_1:56
theorem :: SEQ_1:57
for
seq1,
seq,
seq1' being
Real_Sequence holds
(
(seq1 /" seq) + (seq1' /" seq) = (seq1 + seq1') /" seq &
(seq1 /" seq) - (seq1' /" seq) = (seq1 - seq1') /" seq )
theorem :: SEQ_1:58
theorem :: SEQ_1:59
theorem Th60: :: SEQ_1:60
theorem :: SEQ_1:61
theorem Th62: :: SEQ_1:62
theorem :: SEQ_1:63
theorem :: SEQ_1:64