:: SQUARE_1 semantic presentation

theorem Th1: :: SQUARE_1:1
canceled;

theorem Th2: :: SQUARE_1:2
for x being real number st 1 < x holds
1 / x < 1
proof end;

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
for x, y being real number holds
( not 0 <= x * y or ( 0 <= x & 0 <= y ) or ( x <= 0 & y <= 0 ) ) by ;

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;

scheme :: SQUARE_1:sch 29
s29{ P1[ set ], P2[ set ] } :
ex z being real number st
for x, y being real number st P1[x] & P2[y] holds
( x <= z & z <= y )
provided
E20: for x, y being real number st P1[x] & P2[y] holds
x <= y
proof end;

definition
let x be Element of REAL , y be Element of REAL ;
canceled;
canceled;
redefine func min as min c1,c2 -> Element of REAL ;
coherence
min x,y is Element of REAL
by XREAL_0:def 1;
redefine func max as max c1,c2 -> Element of REAL ;
coherence
max x,y is Element of REAL
by XREAL_0:def 1;
end;

:: 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
for x, y being real number holds min x,y <= x by XXREAL_0:17;

theorem Th36: :: SQUARE_1:36
canceled;

theorem Th37: :: SQUARE_1:37
canceled;

theorem Th38: :: SQUARE_1:38
for x, y being real number holds
( min x,y = x or min x,y = y ) by XXREAL_0:15;

theorem Th39: :: SQUARE_1:39
for x, y, z being real number holds
( ( x <= y & x <= z ) iff x <= min y,z ) by XXREAL_0:20, XXREAL_0:22;

theorem Th40: :: SQUARE_1:40
for x, y, z being real number holds min x,(min y,z) = min (min x,y),z by XXREAL_0:33;

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
for x, y being real number holds x <= max x,y by XXREAL_0:25;

theorem Th47: :: SQUARE_1:47
canceled;

theorem Th48: :: SQUARE_1:48
canceled;

theorem Th49: :: SQUARE_1:49
for x, y being real number holds
( max x,y = x or max x,y = y ) by XXREAL_0:16;

theorem Th50: :: SQUARE_1:50
for y, x, z being real number holds
( ( y <= x & z <= x ) iff max y,z <= x ) by XXREAL_0:28, XXREAL_0:30;

theorem Th51: :: SQUARE_1:51
for x, y, z being real number holds max x,(max y,z) = max (max x,y),z by XXREAL_0:34;

theorem Th52: :: SQUARE_1:52
canceled;

theorem Th53: :: SQUARE_1:53
for x, y being real number holds (min x,y) + (max x,y) = x + y
proof end;

theorem Th54: :: SQUARE_1:54
for x, y being real number holds max x,(min x,y) = x by XXREAL_0:36;

theorem Th55: :: SQUARE_1:55
for x, y being real number holds min x,(max x,y) = x by XXREAL_0:35;

theorem Th56: :: SQUARE_1:56
for x, y, z being real number holds min x,(max y,z) = max (min x,y),(min x,z) by XXREAL_0:38;

theorem Th57: :: SQUARE_1:57
for x, y, z being real number holds max x,(min y,z) = min (max x,y),(max x,z) by XXREAL_0:39;

definition
let x be complex number ;
func c1 ^2 -> set equals :: SQUARE_1:def 3
x * x;
correctness
coherence
x * x is set
;
;
end;

:: deftheorem Def3 defines ^2 SQUARE_1:def 3 :
for x being complex number holds x ^2 = x * x;

registration
let x be complex number ;
cluster a1 ^2 -> complex ;
coherence
x ^2 is complex
;
end;

registration
let x be real number ;
cluster a1 ^2 -> complex real ;
coherence
x ^2 is real
;
end;

definition
let x be Element of COMPLEX ;
redefine func ^2 as c1 ^2 -> Element of COMPLEX ;
coherence
x ^2 is Element of COMPLEX
by XCMPLX_0:def 2;
end;

definition
let x be Element of REAL ;
redefine func ^2 as c1 ^2 -> Element of REAL ;
coherence
x ^2 is Element of REAL
by XREAL_0:def 1;
end;

theorem Th58: :: SQUARE_1:58
canceled;

theorem Th59: :: SQUARE_1:59
1 ^2 = 1 ;

theorem Th60: :: SQUARE_1:60
0 ^2 = 0 ;

theorem Th61: :: SQUARE_1:61
for a being complex number holds a ^2 = (- a) ^2 ;

theorem Th62: :: SQUARE_1:62
canceled;

theorem Th63: :: SQUARE_1:63
for a, b being complex number holds (a + b) ^2 = ((a ^2 ) + ((2 * a) * b)) + (b ^2 ) ;

theorem Th64: :: SQUARE_1:64
for a, b being complex number holds (a - b) ^2 = ((a ^2 ) - ((2 * a) * b)) + (b ^2 ) ;

theorem Th65: :: SQUARE_1:65
for a being complex number holds (a + 1) ^2 = ((a ^2 ) + (2 * a)) + 1 ;

theorem Th66: :: SQUARE_1:66
for a being complex number holds (a - 1) ^2 = ((a ^2 ) - (2 * a)) + 1 ;

theorem Th67: :: SQUARE_1:67
for a, b being complex number holds (a - b) * (a + b) = (a ^2 ) - (b ^2 ) ;

theorem Th68: :: SQUARE_1:68
for a, b being complex number holds (a * b) ^2 = (a ^2 ) * (b ^2 ) ;

theorem Th69: :: SQUARE_1:69
for a, b being complex number holds (a / b) ^2 = (a ^2 ) / (b ^2 ) by XCMPLX_1:77;

theorem Th70: :: SQUARE_1:70
for a, b being complex number st (a ^2 ) - (b ^2 ) <> 0 holds
1 / (a + b) = (a - b) / ((a ^2 ) - (b ^2 ))
proof end;

theorem Th71: :: SQUARE_1:71
for a, b being complex number st (a ^2 ) - (b ^2 ) <> 0 holds
1 / (a - b) = (a + b) / ((a ^2 ) - (b ^2 ))
proof end;

theorem Th72: :: SQUARE_1:72
for a being real number holds 0 <= a ^2 by XREAL_1:65;

theorem Th73: :: SQUARE_1:73
for a being complex number st a ^2 = 0 holds
a = 0 by XCMPLX_1:6;

theorem Th74: :: SQUARE_1:74
for a being real number st 0 <> a holds
0 < a ^2
proof end;

theorem Th75: :: SQUARE_1:75
for a being real number st 0 < a & a < 1 holds
a ^2 < a
proof end;

theorem Th76: :: SQUARE_1:76
for a being real number st 1 < a holds
a < a ^2
proof end;

Lemma46: for a being real number st 0 < a holds
ex x being real number st
( 0 < x & x ^2 < a )
proof end;

theorem Th77: :: SQUARE_1:77
for x, y being real number st 0 <= x & x <= y holds
x ^2 <= y ^2
proof end;

theorem Th78: :: SQUARE_1:78
for x, y being real number st 0 <= x & x < y holds
x ^2 < y ^2
proof end;

Lemma49: for x, y being real number st 0 <= x & 0 <= y & x ^2 = y ^2 holds
x = y
proof end;

definition
let a be real number ;
assume E20: 0 <= a ;
func sqrt c1 -> real number means :Def4: :: SQUARE_1:def 4
( 0 <= it & it ^2 = a );
existence
ex b1 being real number st
( 0 <= b1 & b1 ^2 = a )
proof end;
uniqueness
for b1, b2 being real number st 0 <= b1 & b1 ^2 = a & 0 <= b2 & b2 ^2 = a holds
b1 = b2
by ;
end;

:: deftheorem Def4 defines sqrt SQUARE_1:def 4 :
for a being real number st 0 <= a holds
for b2 being real number holds
( b2 = sqrt a iff ( 0 <= b2 & b2 ^2 = a ) );

definition
let a be Element of REAL ;
redefine func sqrt as sqrt c1 -> Element of REAL ;
coherence
sqrt a is Element of REAL
by XREAL_0:def 1;
end;

theorem Th79: :: SQUARE_1:79
canceled;

theorem Th80: :: SQUARE_1:80
canceled;

theorem Th81: :: SQUARE_1:81
canceled;

theorem Th82: :: SQUARE_1:82
sqrt 0 = 0 by Lemma22, ;

theorem Th83: :: SQUARE_1:83
sqrt 1 = 1 by Lemma22, ;

Lemma86: for a being real number st 0 <= a holds
sqrt (a ^2 ) = a
proof end;

Lemma87: for x, y being real number st 0 <= x & x < y holds
sqrt x < sqrt y
proof end;

theorem Th84: :: SQUARE_1:84
1 < sqrt 2 by Lemma25, Lemma23;

Lemma88: 2 ^2 = 2 * 2
;

theorem Th85: :: SQUARE_1:85
sqrt 4 = 2 by Lemma22, Lemma26;

theorem Th86: :: SQUARE_1:86
sqrt 2 < 2 by Lemma25, Lemma27;

theorem Th87: :: SQUARE_1:87
canceled;

theorem Th88: :: SQUARE_1:88
canceled;

theorem Th89: :: SQUARE_1:89
for a being real number st 0 <= a holds
sqrt (a ^2 ) = a by Lemma24;

theorem Th90: :: SQUARE_1:90
for a being real number st a <= 0 holds
sqrt (a ^2 ) = - a
proof end;

theorem Th91: :: SQUARE_1:91
canceled;

theorem Th92: :: SQUARE_1:92
for a being real number st 0 <= a & sqrt a = 0 holds
a = 0 by Lemma22, ;

theorem Th93: :: SQUARE_1:93
for a being real number st 0 < a holds
0 < sqrt a
proof end;

theorem Th94: :: SQUARE_1:94
for x, y being real number st 0 <= x & x <= y holds
sqrt x <= sqrt y
proof end;

theorem Th95: :: SQUARE_1:95
for x, y being real number st 0 <= x & x < y holds
sqrt x < sqrt y by Lemma25;

theorem Th96: :: SQUARE_1:96
for x, y being real number st 0 <= x & 0 <= y & sqrt x = sqrt y holds
x = y
proof end;

theorem Th97: :: SQUARE_1:97
for a, b being real number st 0 <= a & 0 <= b holds
sqrt (a * b) = (sqrt a) * (sqrt b)
proof end;

theorem Th98: :: SQUARE_1:98
canceled;

theorem Th99: :: SQUARE_1:99
for a, b being real number st 0 <= a & 0 <= b holds
sqrt (a / b) = (sqrt a) / (sqrt b)
proof end;

theorem Th100: :: SQUARE_1:100
canceled;

theorem Th101: :: SQUARE_1:101
for a being real number st 0 < a holds
sqrt (1 / a) = 1 / (sqrt a) by Lemma23, ;

theorem Th102: :: SQUARE_1:102
for a being real number st 0 < a holds
(sqrt a) / a = 1 / (sqrt a)
proof end;

theorem Th103: :: SQUARE_1:103
for a being real number st 0 < a holds
a / (sqrt a) = sqrt a
proof end;

theorem Th104: :: SQUARE_1:104
for a, b being real number st 0 <= a & 0 <= b holds
((sqrt a) - (sqrt b)) * ((sqrt a) + (sqrt b)) = a - b
proof end;

Lemma92: for a, b being real number st 0 <= a & 0 <= b & a <> b holds
((sqrt a) ^2 ) - ((sqrt b) ^2 ) <> 0
proof end;

theorem Th105: :: SQUARE_1:105
for a, b being real number st 0 <= a & 0 <= b & a <> b holds
1 / ((sqrt a) + (sqrt b)) = ((sqrt a) - (sqrt b)) / (a - b)
proof end;

theorem Th106: :: SQUARE_1:106
for b, a being real number st 0 <= b & b < a holds
1 / ((sqrt a) + (sqrt b)) = ((sqrt a) - (sqrt b)) / (a - b)
proof end;

theorem Th107: :: SQUARE_1:107
for a, b being real number st 0 <= a & 0 <= b holds
1 / ((sqrt a) - (sqrt b)) = ((sqrt a) + (sqrt b)) / (a - b)
proof end;

theorem Th108: :: SQUARE_1:108
for b, a being real number st 0 <= b & b < a holds
1 / ((sqrt a) - (sqrt b)) = ((sqrt a) + (sqrt b)) / (a - b)
proof end;

theorem Th109: :: SQUARE_1:109
for x, y being complex number holds
( not x ^2 = y ^2 or x = y or x = - y )
proof end;

theorem Th110: :: SQUARE_1:110
for x being complex number holds
( not x ^2 = 1 or x = 1 or x = - 1 )
proof end;

theorem Th111: :: SQUARE_1:111
for x being real number st 0 <= x & x <= 1 holds
x ^2 <= x
proof end;

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
for x being real number st (x ^2 ) - 1 <= 0 holds
( - 1 <= x & x <= 1 )
proof end;

theorem Th113: :: SQUARE_1:113
for x, y, z being real number holds
( ( x < y & x < z ) iff x < min y,z ) by XXREAL_0:21, XXREAL_0:23;

theorem Th114: :: SQUARE_1:114
for a, x being real number st a <= 0 & x < a holds
x ^2 > a ^2
proof end;