:: SEQ_4 semantic presentation
Lm1:
for r1, r, g being real number st 0 < r1 & r1 <= r & 0 <= g holds
g / r <= g / r1
by XREAL_1:120;
Lm2:
for s being real number st 0 < s holds
0 < s / 3
by XREAL_1:224;
Lm3:
for g, r, g1, r1 being real number st 0 < g & 0 <= r & g <= g1 & r < r1 holds
g * r < g1 * r1
by XREAL_1:99;
Lm4:
for g, r, g1, r1 being real number st 0 <= g & 0 <= r & g <= g1 & r <= r1 holds
g * r <= g1 * r1
by XREAL_1:68;
theorem :: SEQ_4:1
canceled;
theorem :: SEQ_4:2
canceled;
theorem :: SEQ_4:3
canceled;
theorem :: SEQ_4:4
canceled;
theorem :: SEQ_4:5
canceled;
theorem :: SEQ_4:6
canceled;
theorem :: SEQ_4:7
canceled;
theorem Th8: :: SEQ_4:8
theorem Th9: :: SEQ_4:9
theorem Th10: :: SEQ_4:10
:: deftheorem Def1 defines bounded_above SEQ_4:def 1 :
:: deftheorem Def2 defines bounded_below SEQ_4:def 2 :
:: deftheorem Def3 defines bounded SEQ_4:def 3 :
theorem :: SEQ_4:11
canceled;
theorem :: SEQ_4:12
canceled;
theorem :: SEQ_4:13
canceled;
theorem Th14: :: SEQ_4:14
theorem Th15: :: SEQ_4:15
theorem Th16: :: SEQ_4:16
theorem Th17: :: SEQ_4:17
theorem Th18: :: SEQ_4:18
theorem Th19: :: SEQ_4:19
:: deftheorem Def4 defines upper_bound SEQ_4:def 4 :
:: deftheorem Def5 defines lower_bound SEQ_4:def 5 :
theorem :: SEQ_4:20
canceled;
theorem :: SEQ_4:21
canceled;
theorem Th22: :: SEQ_4:22
theorem Th23: :: SEQ_4:23
theorem :: SEQ_4:24
theorem :: SEQ_4:25
theorem Th26: :: SEQ_4:26
theorem :: SEQ_4:27
theorem :: SEQ_4:28
theorem Th29: :: SEQ_4:29
theorem Th30: :: SEQ_4:30
theorem Th31: :: SEQ_4:31
theorem :: SEQ_4:32
theorem Th33: :: SEQ_4:33
theorem :: SEQ_4:34
canceled;
theorem Th35: :: SEQ_4:35
theorem :: SEQ_4:36
theorem Th37: :: SEQ_4:37
theorem :: SEQ_4:38
theorem Th39: :: SEQ_4:39
theorem Th40: :: SEQ_4:40
theorem :: SEQ_4:41
theorem :: SEQ_4:42
theorem Th43: :: SEQ_4:43
theorem Th44: :: SEQ_4:44
theorem :: SEQ_4:45
theorem :: SEQ_4:46
theorem Th47: :: SEQ_4:47
theorem Th48: :: SEQ_4:48
theorem :: SEQ_4:49
theorem :: SEQ_4:50
theorem Th51: :: SEQ_4:51
theorem Th52: :: SEQ_4:52
theorem Th53: :: SEQ_4:53
theorem :: SEQ_4:54
theorem :: SEQ_4:55
theorem Th56: :: SEQ_4:56
theorem Th57: :: SEQ_4:57
theorem :: SEQ_4:58
theorem :: SEQ_4:59
theorem Th60: :: SEQ_4:60
theorem :: SEQ_4:61
theorem Th62: :: SEQ_4:62
theorem :: SEQ_4:63
theorem :: SEQ_4:64
theorem :: SEQ_4:65