:: SQUARE_1 semantic presentation
theorem Th1: :: SQUARE_1:1
canceled;
theorem Th2: :: SQUARE_1:2
Lemma22:
for x, y being real number st x < y holds
0 < y - x
by XREAL_1:52;
Lemma23:
for x, y being real number st x <= y holds
0 <= y - x
by XREAL_1:50;
Lemma24:
for x, y being real number st 0 <= x & 0 <= y holds
0 <= x * y
by XREAL_1:129;
Lemma25:
for x, y being real number st 0 < x & 0 < y holds
0 < x * y
by XREAL_1:131;
Lemma26:
for x, y being real number st 0 < x & y < 0 holds
x * y < 0
by XREAL_1:134;
theorem Th3: :: SQUARE_1:3
canceled;
theorem Th4: :: SQUARE_1:4
canceled;
theorem Th5: :: SQUARE_1:5
canceled;
theorem Th6: :: SQUARE_1:6
canceled;
theorem Th7: :: SQUARE_1:7
canceled;
theorem Th8: :: SQUARE_1:8
canceled;
theorem Th9: :: SQUARE_1:9
canceled;
theorem Th10: :: SQUARE_1:10
canceled;
theorem Th11: :: SQUARE_1:11
canceled;
theorem Th12: :: SQUARE_1:12
canceled;
theorem Th13: :: SQUARE_1:13
canceled;
theorem Th14: :: SQUARE_1:14
canceled;
theorem Th15: :: SQUARE_1:15
canceled;
theorem Th16: :: SQUARE_1:16
canceled;
theorem Th17: :: SQUARE_1:17
canceled;
theorem Th18: :: SQUARE_1:18
canceled;
theorem Th19: :: SQUARE_1:19
canceled;
theorem Th20: :: SQUARE_1:20
canceled;
theorem Th21: :: SQUARE_1:21
canceled;
theorem Th22: :: SQUARE_1:22
canceled;
theorem Th23: :: SQUARE_1:23
canceled;
theorem Th24: :: SQUARE_1:24
canceled;
theorem Th25: :: SQUARE_1:25
Lemma27:
for a, b being real number st 0 <= a & 0 <= b holds
0 <= a / b
by XREAL_1:138;
Lemma28:
for x, y being real number st 0 < x holds
y - x < y
by XREAL_1:46;
:: deftheorem Def1 SQUARE_1:def 1 :
canceled;
:: deftheorem Def2 SQUARE_1:def 2 :
canceled;
theorem Th26: :: SQUARE_1:26
canceled;
theorem Th27: :: SQUARE_1:27
canceled;
theorem Th28: :: SQUARE_1:28
canceled;
theorem Th29: :: SQUARE_1:29
canceled;
theorem Th30: :: SQUARE_1:30
canceled;
theorem Th31: :: SQUARE_1:31
canceled;
theorem Th32: :: SQUARE_1:32
canceled;
theorem Th33: :: SQUARE_1:33
canceled;
theorem Th34: :: SQUARE_1:34
canceled;
theorem Th35: :: SQUARE_1:35
theorem Th36: :: SQUARE_1:36
canceled;
theorem Th37: :: SQUARE_1:37
canceled;
theorem Th38: :: SQUARE_1:38
theorem Th39: :: SQUARE_1:39
theorem Th40: :: SQUARE_1:40
theorem Th41: :: SQUARE_1:41
canceled;
theorem Th42: :: SQUARE_1:42
canceled;
theorem Th43: :: SQUARE_1:43
canceled;
theorem Th44: :: SQUARE_1:44
canceled;
theorem Th45: :: SQUARE_1:45
canceled;
theorem Th46: :: SQUARE_1:46
theorem Th47: :: SQUARE_1:47
canceled;
theorem Th48: :: SQUARE_1:48
canceled;
theorem Th49: :: SQUARE_1:49
theorem Th50: :: SQUARE_1:50
theorem Th51: :: SQUARE_1:51
theorem Th52: :: SQUARE_1:52
canceled;
theorem Th53: :: SQUARE_1:53
theorem Th54: :: SQUARE_1:54
theorem Th55: :: SQUARE_1:55
theorem Th56: :: SQUARE_1:56
theorem Th57: :: SQUARE_1:57
:: deftheorem Def3 defines ^2 SQUARE_1:def 3 :
theorem Th58: :: SQUARE_1:58
canceled;
theorem Th59: :: SQUARE_1:59
theorem Th60: :: SQUARE_1:60
theorem Th61: :: SQUARE_1:61
theorem Th62: :: SQUARE_1:62
canceled;
theorem Th63: :: SQUARE_1:63
theorem Th64: :: SQUARE_1:64
theorem Th65: :: SQUARE_1:65
theorem Th66: :: SQUARE_1:66
theorem Th67: :: SQUARE_1:67
theorem Th68: :: SQUARE_1:68
theorem Th69: :: SQUARE_1:69
theorem Th70: :: SQUARE_1:70
theorem Th71: :: SQUARE_1:71
theorem Th72: :: SQUARE_1:72
theorem Th73: :: SQUARE_1:73
theorem Th74: :: SQUARE_1:74
theorem Th75: :: SQUARE_1:75
theorem Th76: :: SQUARE_1:76
Lemma46:
for a being real number st 0 < a holds
ex x being real number st
( 0 < x & x ^2 < a )
theorem Th77: :: SQUARE_1:77
theorem Th78: :: SQUARE_1:78
Lemma49:
for x, y being real number st 0 <= x & 0 <= y & x ^2 = y ^2 holds
x = y
:: deftheorem Def4 defines sqrt SQUARE_1:def 4 :
theorem Th79: :: SQUARE_1:79
canceled;
theorem Th80: :: SQUARE_1:80
canceled;
theorem Th81: :: SQUARE_1:81
canceled;
theorem Th82: :: SQUARE_1:82
theorem Th83: :: SQUARE_1:83
Lemma86:
for a being real number st 0 <= a holds
sqrt (a ^2 ) = a
Lemma87:
for x, y being real number st 0 <= x & x < y holds
sqrt x < sqrt y
theorem Th84: :: SQUARE_1:84
Lemma88:
2 ^2 = 2 * 2
;
theorem Th85: :: SQUARE_1:85
theorem Th86: :: SQUARE_1:86
theorem Th87: :: SQUARE_1:87
canceled;
theorem Th88: :: SQUARE_1:88
canceled;
theorem Th89: :: SQUARE_1:89
theorem Th90: :: SQUARE_1:90
theorem Th91: :: SQUARE_1:91
canceled;
theorem Th92: :: SQUARE_1:92
theorem Th93: :: SQUARE_1:93
theorem Th94: :: SQUARE_1:94
theorem Th95: :: SQUARE_1:95
theorem Th96: :: SQUARE_1:96
theorem Th97: :: SQUARE_1:97
theorem Th98: :: SQUARE_1:98
canceled;
theorem Th99: :: SQUARE_1:99
theorem Th100: :: SQUARE_1:100
canceled;
theorem Th101: :: SQUARE_1:101
theorem Th102: :: SQUARE_1:102
theorem Th103: :: SQUARE_1:103
theorem Th104: :: SQUARE_1:104
Lemma92:
for a, b being real number st 0 <= a & 0 <= b & a <> b holds
((sqrt a) ^2 ) - ((sqrt b) ^2 ) <> 0
theorem Th105: :: SQUARE_1:105
theorem Th106: :: SQUARE_1:106
theorem Th107: :: SQUARE_1:107
theorem Th108: :: SQUARE_1:108
theorem Th109: :: SQUARE_1:109
theorem Th110: :: SQUARE_1:110
theorem Th111: :: SQUARE_1:111
Lemma93:
for a, x being real number st a >= 0 & (x - a) * (x + a) <= 0 holds
( - a <= x & x <= a )
by XREAL_1:95;
theorem Th112: :: SQUARE_1:112
theorem Th113: :: SQUARE_1:113
theorem Th114: :: SQUARE_1:114