:: COMPLEX2 semantic presentation

Lemma1: 0. F_Complex = 0
by COMPLFLD:9;

theorem Th1: :: COMPLEX2:1
for b1, b2 being Real holds - (b1 + (b2 * <i> )) = (- b1) + ((- b2) * <i> ) ;

theorem Th2: :: COMPLEX2:2
for b1, b2 being real number holds
not ( b2 > 0 & ( for b3 being real number holds
not ( b3 = (b2 * (- [\(b1 / b2)/])) + b1 & 0 <= b3 & b3 < b2 ) ) )
proof end;

theorem Th3: :: COMPLEX2:3
for b1, b2, b3 being real number holds
( b1 > 0 & b2 >= 0 & b3 >= 0 & b2 < b1 & b3 < b1 implies for b4 being Integer holds
( b2 = b3 + (b1 * b4) implies b2 = b3 ) )
proof end;

theorem Th4: :: COMPLEX2:4
for b1, b2 being real number holds
( sin (b1 - b2) = ((sin b1) * (cos b2)) - ((cos b1) * (sin b2)) & cos (b1 - b2) = ((cos b1) * (cos b2)) + ((sin b1) * (sin b2)) )
proof end;

theorem Th5: :: COMPLEX2:5
for b1 being real number holds
( sin . (b1 - PI ) = - (sin . b1) & cos . (b1 - PI ) = - (cos . b1) )
proof end;

theorem Th6: :: COMPLEX2:6
for b1 being real number holds
( sin (b1 - PI ) = - (sin b1) & cos (b1 - PI ) = - (cos b1) )
proof end;

theorem Th7: :: COMPLEX2:7
for b1, b2 being real number holds
( b1 in ].0,(PI / 2).[ & b2 in ].0,(PI / 2).[ implies ( not ( b1 < b2 & not sin b1 < sin b2 ) & not ( sin b1 < sin b2 & not b1 < b2 ) ) )
proof end;

theorem Th8: :: COMPLEX2:8
for b1, b2 being real number holds
( b1 in ].(PI / 2),PI .[ & b2 in ].(PI / 2),PI .[ implies ( not ( b1 < b2 & not sin b1 > sin b2 ) & not ( sin b1 > sin b2 & not b1 < b2 ) ) )
proof end;

theorem Th9: :: COMPLEX2:9
for b1 being real number
for b2 being Integer holds sin b1 = sin (((2 * PI ) * b2) + b1)
proof end;

theorem Th10: :: COMPLEX2:10
for b1 being real number
for b2 being Integer holds cos b1 = cos (((2 * PI ) * b2) + b1)
proof end;

theorem Th11: :: COMPLEX2:11
for b1 being real number holds
not ( sin b1 = 0 & not cos b1 <> 0 )
proof end;

theorem Th12: :: COMPLEX2:12
for b1, b2 being real number holds
( 0 <= b1 & b1 < 2 * PI & 0 <= b2 & b2 < 2 * PI & sin b1 = sin b2 & cos b1 = cos b2 implies b1 = b2 )
proof end;

definition
let c1 be Element of COMPLEX ;
func F_tize c1 -> Element of F_Complex equals :: COMPLEX2:def 1
a1;
correctness
coherence
c1 is Element of F_Complex
;
by COMPLFLD:def 1;
end;

:: deftheorem Def1 defines F_tize COMPLEX2:def 1 :
for b1 being Element of COMPLEX holds F_tize b1 = b1;

theorem Th13: :: COMPLEX2:13
canceled;

theorem Th14: :: COMPLEX2:14
for b1, b2 being Element of COMPLEX holds F_tize (b1 + b2) = (F_tize b1) + (F_tize b2) ;

theorem Th15: :: COMPLEX2:15
for b1 being Element of COMPLEX holds
( b1 = 0 iff F_tize b1 = 0. F_Complex ) by COMPLFLD:9;

Lemma12: for b1 being complex number holds
( 0 <= Arg b1 & Arg b1 < 2 * PI )
by COMPTRIG:52;

Lemma13: 0c = [*0,0*]
by ARYTM_0:def 7;

theorem Th16: :: COMPLEX2:16
canceled;

theorem Th17: :: COMPLEX2:17
canceled;

theorem Th18: :: COMPLEX2:18
canceled;

theorem Th19: :: COMPLEX2:19
for b1 being complex number holds b1 = [*(|.b1.| * (cos (Arg b1))),(|.b1.| * (sin (Arg b1)))*]
proof end;

Lemma15: [**0,0**] = 0 + (0 * <i> )
by HAHNBAN1:def 1;

theorem Th20: :: COMPLEX2:20
Arg 0 = 0 by Lemma15, COMPTRIG:53;

theorem Th21: :: COMPLEX2:21
for b1 being complex number
for b2 being Real holds
( b1 <> 0 & b1 = [*(|.b1.| * (cos b2)),(|.b1.| * (sin b2))*] & 0 <= b2 & b2 < 2 * PI implies b2 = Arg b1 )
proof end;

theorem Th22: :: COMPLEX2:22
for b1 being complex number holds
( b1 <> 0 implies ( ( Arg b1 < PI implies Arg (- b1) = (Arg b1) + PI ) & ( Arg b1 >= PI implies Arg (- b1) = (Arg b1) - PI ) ) )
proof end;

theorem Th23: :: COMPLEX2:23
for b1 being Real holds
( b1 >= 0 implies Arg [*b1,0*] = 0 )
proof end;

theorem Th24: :: COMPLEX2:24
for b1 being complex number holds
( Arg b1 = 0 iff b1 = [*|.b1.|,0*] )
proof end;

theorem Th25: :: COMPLEX2:25
for b1 being complex number holds
( b1 <> 0 implies ( Arg b1 < PI iff Arg (- b1) >= PI ) )
proof end;

theorem Th26: :: COMPLEX2:26
for b1, b2 being complex number holds
( not ( not b1 <> b2 & not b1 - b2 <> 0 ) implies ( Arg (b1 - b2) < PI iff Arg (b2 - b1) >= PI ) )
proof end;

theorem Th27: :: COMPLEX2:27
for b1 being complex number holds
( Arg b1 in ].0,PI .[ iff Im b1 > 0 )
proof end;

theorem Th28: :: COMPLEX2:28
for b1 being complex number holds
( Arg b1 <> 0 implies ( not ( Arg b1 < PI & not sin (Arg b1) > 0 ) & not ( sin (Arg b1) > 0 & not Arg b1 < PI ) ) )
proof end;

theorem Th29: :: COMPLEX2:29
for b1, b2 being complex number holds
not ( Arg b1 < PI & Arg b2 < PI & not Arg (b1 + b2) < PI )
proof end;

theorem Th30: :: COMPLEX2:30
for b1 being Real holds
( b1 > 0 implies Arg [*0,b1*] = PI / 2 )
proof end;

Lemma25: for b1 being complex number holds
( Arg b1 in ].0,(PI / 2).[ iff ( Re b1 > 0 & Im b1 > 0 ) )
by COMPTRIG:59;

Lemma26: for b1 being complex number holds
( Arg b1 in ].(PI / 2),PI .[ iff ( Re b1 < 0 & Im b1 > 0 ) )
by COMPTRIG:60;

Lemma27: for b1 being complex number holds
not ( Im b1 > 0 & not sin (Arg b1) > 0 )
by COMPTRIG:63;

theorem Th31: :: COMPLEX2:31
canceled;

theorem Th32: :: COMPLEX2:32
canceled;

theorem Th33: :: COMPLEX2:33
canceled;

theorem Th34: :: COMPLEX2:34
for b1 being complex number holds
( Arg b1 = 0 iff ( Re b1 >= 0 & Im b1 = 0 ) )
proof end;

theorem Th35: :: COMPLEX2:35
for b1 being complex number holds
( Arg b1 = PI iff ( Re b1 < 0 & Im b1 = 0 ) )
proof end;

theorem Th36: :: COMPLEX2:36
for b1 being complex number holds
( Im b1 = 0 iff ( Arg b1 = 0 or Arg b1 = PI ) )
proof end;

theorem Th37: :: COMPLEX2:37
for b1 being complex number holds
( Arg b1 <= PI implies Im b1 >= 0 )
proof end;

theorem Th38: :: COMPLEX2:38
for b1 being Element of COMPLEX holds
( b1 <> 0 implies ( cos (Arg (- b1)) = - (cos (Arg b1)) & sin (Arg (- b1)) = - (sin (Arg b1)) ) )
proof end;

theorem Th39: :: COMPLEX2:39
for b1 being complex number holds
( b1 <> 0 implies ( cos (Arg b1) = (Re b1) / |.b1.| & sin (Arg b1) = (Im b1) / |.b1.| ) )
proof end;

theorem Th40: :: COMPLEX2:40
for b1 being complex number
for b2 being Real holds
( b2 > 0 implies Arg (b1 * [*b2,0*]) = Arg b1 )
proof end;

theorem Th41: :: COMPLEX2:41
for b1 being complex number
for b2 being Real holds
( b2 < 0 implies Arg (b1 * [*b2,0*]) = Arg (- b1) )
proof end;

definition
let c1, c2 be complex number ;
canceled;
func c1 .|. c2 -> Element of COMPLEX equals :: COMPLEX2:def 3
a1 * (a2 *' );
correctness
coherence
c1 * (c2 *' ) is Element of COMPLEX
;
by XCMPLX_0:def 2;
end;

:: deftheorem Def2 COMPLEX2:def 2 :
canceled;

:: deftheorem Def3 defines .|. COMPLEX2:def 3 :
for b1, b2 being complex number holds b1 .|. b2 = b1 * (b2 *' );

theorem Th42: :: COMPLEX2:42
for b1, b2 being Element of COMPLEX holds b1 .|. b2 = [*(((Re b1) * (Re b2)) + ((Im b1) * (Im b2))),((- ((Re b1) * (Im b2))) + ((Im b1) * (Re b2)))*]
proof end;

theorem Th43: :: COMPLEX2:43
for b1 being Element of COMPLEX holds
( b1 .|. b1 = [*(((Re b1) * (Re b1)) + ((Im b1) * (Im b1))),0*] & b1 .|. b1 = [*(((Re b1) ^2 ) + ((Im b1) ^2 )),0*] )
proof end;

theorem Th44: :: COMPLEX2:44
for b1 being Element of COMPLEX holds
( b1 .|. b1 = [*(|.b1.| ^2 ),0*] & |.b1.| ^2 = Re (b1 .|. b1) )
proof end;

theorem Th45: :: COMPLEX2:45
for b1, b2 being Element of COMPLEX holds |.(b1 .|. b2).| = |.b1.| * |.b2.|
proof end;

theorem Th46: :: COMPLEX2:46
for b1 being Element of COMPLEX holds
( b1 .|. b1 = 0 implies b1 = 0 )
proof end;

theorem Th47: :: COMPLEX2:47
for b1, b2 being Element of COMPLEX holds b1 .|. b2 = (b2 .|. b1) *'
proof end;

theorem Th48: :: COMPLEX2:48
for b1, b2, b3 being Element of COMPLEX holds (b1 + b2) .|. b3 = (b1 .|. b3) + (b2 .|. b3) ;

theorem Th49: :: COMPLEX2:49
for b1, b2, b3 being Element of COMPLEX holds b1 .|. (b2 + b3) = (b1 .|. b2) + (b1 .|. b3)
proof end;

theorem Th50: :: COMPLEX2:50
for b1, b2, b3 being Element of COMPLEX holds (b1 * b2) .|. b3 = b1 * (b2 .|. b3) ;

theorem Th51: :: COMPLEX2:51
for b1, b2, b3 being Element of COMPLEX holds b1 .|. (b2 * b3) = (b2 *' ) * (b1 .|. b3)
proof end;

theorem Th52: :: COMPLEX2:52
for b1, b2, b3 being Element of COMPLEX holds (b1 * b2) .|. b3 = b2 .|. ((b1 *' ) * b3)
proof end;

theorem Th53: :: COMPLEX2:53
for b1, b2, b3, b4, b5 being Element of COMPLEX holds ((b1 * b2) + (b3 * b4)) .|. b5 = (b1 * (b2 .|. b5)) + (b3 * (b4 .|. b5)) ;

theorem Th54: :: COMPLEX2:54
for b1, b2, b3, b4, b5 being Element of COMPLEX holds b1 .|. ((b2 * b3) + (b4 * b5)) = ((b2 *' ) * (b1 .|. b3)) + ((b4 *' ) * (b1 .|. b5))
proof end;

theorem Th55: :: COMPLEX2:55
for b1, b2 being Element of COMPLEX holds (- b1) .|. b2 = b1 .|. (- b2)
proof end;

theorem Th56: :: COMPLEX2:56
for b1, b2 being Element of COMPLEX holds (- b1) .|. b2 = - (b1 .|. b2) ;

theorem Th57: :: COMPLEX2:57
for b1, b2 being Element of COMPLEX holds - (b1 .|. b2) = b1 .|. (- b2)
proof end;

theorem Th58: :: COMPLEX2:58
for b1, b2 being Element of COMPLEX holds (- b1) .|. (- b2) = b1 .|. b2
proof end;

theorem Th59: :: COMPLEX2:59
for b1, b2, b3 being Element of COMPLEX holds (b1 - b2) .|. b3 = (b1 .|. b3) - (b2 .|. b3) ;

theorem Th60: :: COMPLEX2:60
for b1, b2, b3 being Element of COMPLEX holds b1 .|. (b2 - b3) = (b1 .|. b2) - (b1 .|. b3)
proof end;

theorem Th61: :: COMPLEX2:61
for b1 being Element of COMPLEX holds
( 0 .|. b1 = 0c & b1 .|. 0 = 0 ) by COMPLEX1:113;

theorem Th62: :: COMPLEX2:62
for b1, b2 being Element of COMPLEX holds (b1 + b2) .|. (b1 + b2) = (((b1 .|. b1) + (b1 .|. b2)) + (b2 .|. b1)) + (b2 .|. b2)
proof end;

theorem Th63: :: COMPLEX2:63
for b1, b2 being Element of COMPLEX holds (b1 - b2) .|. (b1 - b2) = (((b1 .|. b1) - (b1 .|. b2)) - (b2 .|. b1)) + (b2 .|. b2)
proof end;

Lemma47: for b1 being Element of COMPLEX holds |.b1.| ^2 = ((Re b1) ^2 ) + ((Im b1) ^2 )
proof end;

theorem Th64: :: COMPLEX2:64
for b1, b2 being Element of COMPLEX holds
( Re (b1 .|. b2) = 0 iff ( Im (b1 .|. b2) = |.b1.| * |.b2.| or Im (b1 .|. b2) = - (|.b1.| * |.b2.|) ) )
proof end;

definition
let c1 be complex number ;
let c2 be Real;
func Rotate c1,c2 -> Element of COMPLEX equals :: COMPLEX2:def 4
[*(|.a1.| * (cos (a2 + (Arg a1)))),(|.a1.| * (sin (a2 + (Arg a1))))*];
correctness
coherence
[*(|.c1.| * (cos (c2 + (Arg c1)))),(|.c1.| * (sin (c2 + (Arg c1))))*] is Element of COMPLEX
;
;
end;

:: deftheorem Def4 defines Rotate COMPLEX2:def 4 :
for b1 being complex number
for b2 being Real holds Rotate b1,b2 = [*(|.b1.| * (cos (b2 + (Arg b1)))),(|.b1.| * (sin (b2 + (Arg b1))))*];

theorem Th65: :: COMPLEX2:65
for b1 being Element of COMPLEX holds Rotate b1,0 = b1 by Th19;

theorem Th66: :: COMPLEX2:66
for b1 being Element of COMPLEX
for b2 being Real holds
( Rotate b1,b2 = 0 iff b1 = 0 )
proof end;

theorem Th67: :: COMPLEX2:67
for b1 being Element of COMPLEX
for b2 being Real holds |.(Rotate b1,b2).| = |.b1.|
proof end;

theorem Th68: :: COMPLEX2:68
for b1 being Element of COMPLEX
for b2 being Real holds
not ( b1 <> 0 & ( for b3 being Integer holds
not Arg (Rotate b1,b2) = ((2 * PI ) * b3) + (b2 + (Arg b1)) ) )
proof end;

theorem Th69: :: COMPLEX2:69
for b1 being Element of COMPLEX holds Rotate b1,(- (Arg b1)) = [*|.b1.|,0*] by SIN_COS:34;

theorem Th70: :: COMPLEX2:70
for b1 being Element of COMPLEX
for b2 being Real holds
( Re (Rotate b1,b2) = ((Re b1) * (cos b2)) - ((Im b1) * (sin b2)) & Im (Rotate b1,b2) = ((Re b1) * (sin b2)) + ((Im b1) * (cos b2)) )
proof end;

theorem Th71: :: COMPLEX2:71
for b1, b2 being Element of COMPLEX
for b3 being Real holds Rotate (b1 + b2),b3 = (Rotate b1,b3) + (Rotate b2,b3)
proof end;

theorem Th72: :: COMPLEX2:72
for b1 being Element of COMPLEX
for b2 being Real holds Rotate (- b1),b2 = - (Rotate b1,b2)
proof end;

theorem Th73: :: COMPLEX2:73
for b1, b2 being Element of COMPLEX
for b3 being Real holds Rotate (b1 - b2),b3 = (Rotate b1,b3) - (Rotate b2,b3)
proof end;

theorem Th74: :: COMPLEX2:74
for b1 being Element of COMPLEX holds Rotate b1,PI = - b1
proof end;

definition
let c1, c2 be Element of COMPLEX ;
func angle c1,c2 -> Real equals :Def5: :: COMPLEX2:def 5
Arg (Rotate a2,(- (Arg a1))) if not ( not Arg a1 = 0 & not a2 <> 0 )
otherwise (2 * PI ) - (Arg a1);
correctness
coherence
( ( not ( not Arg c1 = 0 & not c2 <> 0 ) implies Arg (Rotate c2,(- (Arg c1))) is Real ) & not ( not Arg c1 = 0 & not c2 <> 0 & not (2 * PI ) - (Arg c1) is Real ) )
;
consistency
for b1 being Real holds
verum
;
;
end;

:: deftheorem Def5 defines angle COMPLEX2:def 5 :
for b1, b2 being Element of COMPLEX holds
( ( not ( not Arg b1 = 0 & not b2 <> 0 ) implies angle b1,b2 = Arg (Rotate b2,(- (Arg b1))) ) & not ( not Arg b1 = 0 & not b2 <> 0 & not angle b1,b2 = (2 * PI ) - (Arg b1) ) );

theorem Th75: :: COMPLEX2:75
for b1 being Element of COMPLEX
for b2 being Real holds
( b2 >= 0 implies angle [*b2,0*],b1 = Arg b1 )
proof end;

theorem Th76: :: COMPLEX2:76
for b1, b2 being Element of COMPLEX
for b3 being Real holds
( Arg b1 = Arg b2 & b1 <> 0 & b2 <> 0 implies Arg (Rotate b1,b3) = Arg (Rotate b2,b3) )
proof end;

theorem Th77: :: COMPLEX2:77
for b1, b2 being Element of COMPLEX
for b3 being Real holds
( b3 > 0 implies angle b1,b2 = angle (b1 * [*b3,0*]),(b2 * [*b3,0*]) )
proof end;

theorem Th78: :: COMPLEX2:78
for b1, b2 being Element of COMPLEX holds
( b1 <> 0 & b2 <> 0 & Arg b1 = Arg b2 implies Arg (- b1) = Arg (- b2) )
proof end;

theorem Th79: :: COMPLEX2:79
for b1, b2 being Element of COMPLEX
for b3 being Real holds
( b1 <> 0 & b2 <> 0 implies angle b1,b2 = angle (Rotate b1,b3),(Rotate b2,b3) )
proof end;

theorem Th80: :: COMPLEX2:80
for b1, b2 being Element of COMPLEX
for b3 being Real holds
( b3 < 0 & b1 <> 0 & b2 <> 0 implies angle b1,b2 = angle (b1 * [*b3,0*]),(b2 * [*b3,0*]) )
proof end;

theorem Th81: :: COMPLEX2:81
for b1, b2 being Element of COMPLEX holds
( b1 <> 0 & b2 <> 0 implies angle b1,b2 = angle (- b1),(- b2) )
proof end;

theorem Th82: :: COMPLEX2:82
for b1, b2 being Element of COMPLEX holds
( b1 <> 0 & angle b2,b1 = 0 implies angle b2,(- b1) = PI )
proof end;

theorem Th83: :: COMPLEX2:83
for b1, b2 being Element of COMPLEX holds
( b1 <> 0 & b2 <> 0 implies ( cos (angle b1,b2) = (Re (b1 .|. b2)) / (|.b1.| * |.b2.|) & sin (angle b1,b2) = - ((Im (b1 .|. b2)) / (|.b1.| * |.b2.|)) ) )
proof end;

definition
let c1, c2, c3 be complex number ;
func angle c1,c2,c3 -> real number equals :Def6: :: COMPLEX2:def 6
(Arg (a3 - a2)) - (Arg (a1 - a2)) if (Arg (a3 - a2)) - (Arg (a1 - a2)) >= 0
otherwise (2 * PI ) + ((Arg (a3 - a2)) - (Arg (a1 - a2)));
correctness
coherence
( ( (Arg (c3 - c2)) - (Arg (c1 - c2)) >= 0 implies (Arg (c3 - c2)) - (Arg (c1 - c2)) is real number ) & ( not (Arg (c3 - c2)) - (Arg (c1 - c2)) >= 0 implies (2 * PI ) + ((Arg (c3 - c2)) - (Arg (c1 - c2))) is real number ) )
;
consistency
for b1 being real number holds
verum
;
;
end;

:: deftheorem Def6 defines angle COMPLEX2:def 6 :
for b1, b2, b3 being complex number holds
( ( (Arg (b3 - b2)) - (Arg (b1 - b2)) >= 0 implies angle b1,b2,b3 = (Arg (b3 - b2)) - (Arg (b1 - b2)) ) & ( not (Arg (b3 - b2)) - (Arg (b1 - b2)) >= 0 implies angle b1,b2,b3 = (2 * PI ) + ((Arg (b3 - b2)) - (Arg (b1 - b2))) ) );

theorem Th84: :: COMPLEX2:84
for b1, b2, b3 being Element of COMPLEX holds
( 0 <= angle b1,b2,b3 & angle b1,b2,b3 < 2 * PI )
proof end;

theorem Th85: :: COMPLEX2:85
for b1, b2, b3 being Element of COMPLEX holds angle b1,b2,b3 = angle (b1 - b2),0,(b3 - b2)
proof end;

theorem Th86: :: COMPLEX2:86
for b1, b2, b3, b4 being Element of COMPLEX holds angle b1,b2,b3 = angle (b1 + b4),(b2 + b4),(b3 + b4)
proof end;

theorem Th87: :: COMPLEX2:87
for b1, b2 being Element of COMPLEX holds angle b1,b2 = angle b1,0,b2
proof end;

theorem Th88: :: COMPLEX2:88
for b1, b2, b3 being Element of COMPLEX holds
( angle b1,b2,b3 = 0 implies ( Arg (b1 - b2) = Arg (b3 - b2) & angle b3,b2,b1 = 0 ) )
proof end;

theorem Th89: :: COMPLEX2:89
for b1, b2 being Element of COMPLEX holds
( b1 <> 0 & b2 <> 0 implies ( Re (b1 .|. b2) = 0 iff ( angle b1,0,b2 = PI / 2 or angle b1,0,b2 = (3 / 2) * PI ) ) )
proof end;

theorem Th90: :: COMPLEX2:90
for b1, b2 being Element of COMPLEX holds
( b1 <> 0 & b2 <> 0 implies ( not ( ( Im (b1 .|. b2) = |.b1.| * |.b2.| or Im (b1 .|. b2) = - (|.b1.| * |.b2.|) ) & not angle b1,0,b2 = PI / 2 & not angle b1,0,b2 = (3 / 2) * PI ) & not ( ( angle b1,0,b2 = PI / 2 or angle b1,0,b2 = (3 / 2) * PI ) & not Im (b1 .|. b2) = |.b1.| * |.b2.| & not Im (b1 .|. b2) = - (|.b1.| * |.b2.|) ) ) )
proof end;

Lemma73: for b1, b2, b3 being Element of COMPLEX holds
( b1 <> b2 & b3 <> b2 implies ( Re ((b1 - b2) .|. (b3 - b2)) = 0 iff ( angle b1,b2,b3 = PI / 2 or angle b1,b2,b3 = (3 / 2) * PI ) ) )
proof end;

theorem Th91: :: COMPLEX2:91
for b1, b2, b3 being Element of COMPLEX holds
( b1 <> b2 & b3 <> b2 & ( angle b1,b2,b3 = PI / 2 or angle b1,b2,b3 = (3 / 2) * PI ) implies (|.(b1 - b2).| ^2 ) + (|.(b3 - b2).| ^2 ) = |.(b1 - b3).| ^2 )
proof end;

theorem Th92: :: COMPLEX2:92
for b1, b2, b3 being Element of COMPLEX
for b4 being Real holds
( b1 <> b2 & b2 <> b3 implies angle b1,b2,b3 = angle (Rotate b1,b4),(Rotate b2,b4),(Rotate b3,b4) )
proof end;

theorem Th93: :: COMPLEX2:93
for b1, b2 being Element of COMPLEX holds angle b1,b2,b1 = 0
proof end;

Lemma76: for b1, b2, b3 being Element of COMPLEX holds
( angle b1,b2,b3 <> 0 implies angle b3,b2,b1 = (2 * PI ) - (angle b1,b2,b3) )
proof end;

theorem Th94: :: COMPLEX2:94
for b1, b2, b3 being Element of COMPLEX holds
( angle b1,b2,b3 <> 0 iff (angle b1,b2,b3) + (angle b3,b2,b1) = 2 * PI )
proof end;

theorem Th95: :: COMPLEX2:95
for b1, b2, b3 being Element of COMPLEX holds
not ( angle b1,b2,b3 <> 0 & not angle b3,b2,b1 <> 0 )
proof end;

theorem Th96: :: COMPLEX2:96
for b1, b2, b3 being Element of COMPLEX holds
( angle b1,b2,b3 = PI implies angle b3,b2,b1 = PI )
proof end;

theorem Th97: :: COMPLEX2:97
for b1, b2, b3 being Element of COMPLEX holds
not ( b1 <> b2 & b1 <> b3 & b2 <> b3 & not angle b1,b2,b3 <> 0 & not angle b2,b3,b1 <> 0 & not angle b3,b1,b2 <> 0 )
proof end;

Lemma79: for b1, b2 being Element of COMPLEX holds
( Im b1 = 0 & Re b1 > 0 & 0 < Arg b2 & Arg b2 < PI implies ( ((angle b1,0c ,b2) + (angle 0c ,b2,b1)) + (angle b2,b1,0c ) = PI & 0 < angle 0c ,b2,b1 & 0 < angle b2,b1,0c ) )
proof end;

theorem Th98: :: COMPLEX2:98
for b1, b2, b3 being Element of COMPLEX holds
( b1 <> b2 & b2 <> b3 & 0 < angle b1,b2,b3 & angle b1,b2,b3 < PI implies ( ((angle b1,b2,b3) + (angle b2,b3,b1)) + (angle b3,b1,b2) = PI & 0 < angle b2,b3,b1 & 0 < angle b3,b1,b2 ) )
proof end;

theorem Th99: :: COMPLEX2:99
for b1, b2, b3 being Element of COMPLEX holds
( b1 <> b2 & b2 <> b3 & angle b1,b2,b3 > PI implies ( ((angle b1,b2,b3) + (angle b2,b3,b1)) + (angle b3,b1,b2) = 5 * PI & angle b2,b3,b1 > PI & angle b3,b1,b2 > PI ) )
proof end;

Lemma82: for b1, b2 being Element of COMPLEX holds
( Im b1 = 0 & Re b1 > 0 & Arg b2 = PI implies ( ((angle b1,0,b2) + (angle 0,b2,b1)) + (angle b2,b1,0) = PI & 0 = angle 0,b2,b1 & 0 = angle b2,b1,0 ) )
proof end;

theorem Th100: :: COMPLEX2:100
for b1, b2, b3 being Element of COMPLEX holds
( b1 <> b2 & b2 <> b3 & angle b1,b2,b3 = PI implies ( angle b2,b3,b1 = 0 & angle b3,b1,b2 = 0 ) )
proof end;

theorem Th101: :: COMPLEX2:101
for b1, b2, b3 being Element of COMPLEX holds
not ( b1 <> b2 & b1 <> b3 & b2 <> b3 & angle b1,b2,b3 = 0 & not ( angle b2,b3,b1 = 0 & angle b3,b1,b2 = PI ) & not ( angle b2,b3,b1 = PI & angle b3,b1,b2 = 0 ) )
proof end;

Lemma85: for b1, b2, b3 being Element of COMPLEX holds
( b1 <> b2 & b1 <> b3 & b2 <> b3 & angle b1,b2,b3 = 0 implies (angle b2,b3,b1) + (angle b3,b1,b2) = PI )
proof end;

theorem Th102: :: COMPLEX2:102
for b1, b2, b3 being Element of COMPLEX holds
( ( ((angle b1,b2,b3) + (angle b2,b3,b1)) + (angle b3,b1,b2) = PI or ((angle b1,b2,b3) + (angle b2,b3,b1)) + (angle b3,b1,b2) = 5 * PI ) iff ( b1 <> b2 & b1 <> b3 & b2 <> b3 ) )
proof end;