:: SIN_COS6 semantic presentation
theorem Th1: :: SIN_COS6:1
theorem Th2: :: SIN_COS6:2
theorem Th3: :: SIN_COS6:3
theorem Th4: :: SIN_COS6:4
theorem Th5: :: SIN_COS6:5
theorem Th6: :: SIN_COS6:6
deffunc H1( Integer) -> set = (2 * PI ) * a1;
Lemma42:
dom sin = REAL
by SIN_COS:def 20;
Lemma43:
dom cos = REAL
by SIN_COS:def 22;
Lemma44:
[.((- (PI / 2)) + H1(0)),((PI / 2) + H1(0)).] = [.(- (PI / 2)),(PI / 2).]
;
Lemma45:
[.((PI / 2) + H1(0)),(((3 / 2) * PI ) + H1(0)).] = [.(PI / 2),((3 / 2) * PI ).]
;
Lemma46:
[.H1(0),(PI + H1(0)).] = [.0,PI .]
;
Lemma47:
[.(PI + H1(0)),((2 * PI ) + H1(0)).] = [.PI ,(2 * PI ).]
;
Lemma48:
for r, s being real number st (r ^2 ) + (s ^2 ) = 1 holds
( - 1 <= r & r <= 1 )
Lemma49:
- (PI / 2) < - 0
by XREAL_1:26;
then Lemma50:
- (PI / 2) < PI / 2
;
Lemma51:
PI / 2 < PI / 1
by REAL_2:200;
Lemma52:
1 * PI < (3 / 2) * PI
by XREAL_1:70;
Lemma53:
0 + H1(1) < (PI / 2) + H1(1)
by XREAL_1:8;
Lemma54:
(3 / 2) * PI < 2 * PI
by XREAL_1:70;
Lemma55:
1 * PI < 2 * PI
by XREAL_1:70;
Lemma56:
].(- 1),1.[ c= [.(- 1),1.]
by RCOMP_1:15;
Lemma57:
].(- (PI / 2)),0.[ c= [.(- (PI / 2)),0.]
by RCOMP_1:15;
Lemma58:
].(- (PI / 2)),(PI / 2).[ c= [.(- (PI / 2)),(PI / 2).]
by RCOMP_1:15;
Lemma59:
].0,(PI / 2).[ c= [.0,(PI / 2).]
by RCOMP_1:15;
Lemma60:
].0,PI .[ c= [.0,PI .]
by RCOMP_1:15;
Lemma61:
].(PI / 2),PI .[ c= [.(PI / 2),PI .]
by RCOMP_1:15;
Lemma62:
].(PI / 2),((3 / 2) * PI ).[ c= [.(PI / 2),((3 / 2) * PI ).]
by RCOMP_1:15;
Lemma63:
].PI ,((3 / 2) * PI ).[ c= [.PI ,((3 / 2) * PI ).]
by RCOMP_1:15;
Lemma64:
].PI ,(2 * PI ).[ c= [.PI ,(2 * PI ).]
by RCOMP_1:15;
Lemma65:
].((3 / 2) * PI ),(2 * PI ).[ c= [.((3 / 2) * PI ),(2 * PI ).]
by RCOMP_1:15;
Lemma66:
[.(- (PI / 2)),0.] c= [.(- (PI / 2)),(PI / 2).]
by TREAL_1:1;
Lemma67:
[.0,(PI / 2).] c= [.(- (PI / 2)),(PI / 2).]
by , TREAL_1:1;
Lemma68:
[.0,(PI / 2).] c= [.0,PI .]
by , TREAL_1:1;
Lemma69:
[.(PI / 2),PI .] c= [.(PI / 2),((3 / 2) * PI ).]
by , TREAL_1:1;
Lemma70:
[.(PI / 2),PI .] c= [.0,PI .]
by TREAL_1:1;
Lemma71:
[.PI ,((3 / 2) * PI ).] c= [.(PI / 2),((3 / 2) * PI ).]
by , TREAL_1:1;
Lemma72:
[.PI ,((3 / 2) * PI ).] c= [.PI ,(2 * PI ).]
by , TREAL_1:1;
Lemma73:
[.((3 / 2) * PI ),(2 * PI ).] c= [.((- (PI / 2)) + H1(1)),((PI / 2) + H1(1)).]
by , TREAL_1:1;
Lemma74:
[.((3 / 2) * PI ),(2 * PI ).] c= [.PI ,(2 * PI ).]
by , TREAL_1:1;
theorem Th7: :: SIN_COS6:7
theorem Th8: :: SIN_COS6:8
theorem Th9: :: SIN_COS6:9
theorem Th10: :: SIN_COS6:10
theorem Th11: :: SIN_COS6:11
theorem Th12: :: SIN_COS6:12
theorem Th13: :: SIN_COS6:13
theorem Th14: :: SIN_COS6:14
theorem Th15: :: SIN_COS6:15
theorem Th16: :: SIN_COS6:16
theorem Th17: :: SIN_COS6:17
theorem Th18: :: SIN_COS6:18
theorem Th19: :: SIN_COS6:19
theorem Th20: :: SIN_COS6:20
theorem Th21: :: SIN_COS6:21
theorem Th22: :: SIN_COS6:22
theorem Th23: :: SIN_COS6:23
theorem Th24: :: SIN_COS6:24
theorem Th25: :: SIN_COS6:25
theorem Th26: :: SIN_COS6:26
theorem Th27: :: SIN_COS6:27
theorem Th28: :: SIN_COS6:28
theorem Th29: :: SIN_COS6:29
theorem Th30: :: SIN_COS6:30
theorem Th31: :: SIN_COS6:31
theorem Th32: :: SIN_COS6:32
theorem Th33: :: SIN_COS6:33
theorem Th34: :: SIN_COS6:34
theorem Th35: :: SIN_COS6:35
theorem Th36: :: SIN_COS6:36
theorem Th37: :: SIN_COS6:37
theorem Th38: :: SIN_COS6:38
theorem Th39: :: SIN_COS6:39
theorem Th40: :: SIN_COS6:40
theorem Th41: :: SIN_COS6:41
theorem Th42: :: SIN_COS6:42
theorem Th43: :: SIN_COS6:43
theorem Th44: :: SIN_COS6:44
theorem Th45: :: SIN_COS6:45
theorem Th46: :: SIN_COS6:46
theorem Th47: :: SIN_COS6:47
theorem Th48: :: SIN_COS6:48
theorem Th49: :: SIN_COS6:49
theorem Th50: :: SIN_COS6:50
theorem Th51: :: SIN_COS6:51
theorem Th52: :: SIN_COS6:52
theorem Th53: :: SIN_COS6:53
theorem Th54: :: SIN_COS6:54
theorem Th55: :: SIN_COS6:55
theorem Th56: :: SIN_COS6:56
theorem Th57: :: SIN_COS6:57
theorem Th58: :: SIN_COS6:58
registration
cluster sin | [.(- (PI / 2)),0.] -> one-to-one ;
coherence
sin | [.(- (PI / 2)),0.] is one-to-one
cluster sin | [.0,(PI / 2).] -> one-to-one ;
coherence
sin | [.0,(PI / 2).] is one-to-one
cluster sin | [.(PI / 2),PI .] -> one-to-one ;
coherence
sin | [.(PI / 2),PI .] is one-to-one
cluster sin | [.PI ,((3 / 2) * PI ).] -> one-to-one ;
coherence
sin | [.PI ,((3 / 2) * PI ).] is one-to-one
cluster sin | [.((3 / 2) * PI ),(2 * PI ).] -> one-to-one ;
coherence
sin | [.((3 / 2) * PI ),(2 * PI ).] is one-to-one
end;
registration
cluster sin | ].(- (PI / 2)),(PI / 2).[ -> one-to-one ;
coherence
sin | ].(- (PI / 2)),(PI / 2).[ is one-to-one
cluster sin | ].(PI / 2),((3 / 2) * PI ).[ -> one-to-one ;
coherence
sin | ].(PI / 2),((3 / 2) * PI ).[ is one-to-one
cluster sin | ].(- (PI / 2)),0.[ -> one-to-one ;
coherence
sin | ].(- (PI / 2)),0.[ is one-to-one
cluster sin | ].0,(PI / 2).[ -> one-to-one ;
coherence
sin | ].0,(PI / 2).[ is one-to-one
cluster sin | ].(PI / 2),PI .[ -> one-to-one ;
coherence
sin | ].(PI / 2),PI .[ is one-to-one
cluster sin | ].PI ,((3 / 2) * PI ).[ -> one-to-one ;
coherence
sin | ].PI ,((3 / 2) * PI ).[ is one-to-one
cluster sin | ].((3 / 2) * PI ),(2 * PI ).[ -> one-to-one ;
coherence
sin | ].((3 / 2) * PI ),(2 * PI ).[ is one-to-one
end;
theorem Th59: :: SIN_COS6:59
theorem Th60: :: SIN_COS6:60
registration
cluster cos | ].0,PI .[ -> one-to-one ;
coherence
cos | ].0,PI .[ is one-to-one
cluster cos | ].PI ,(2 * PI ).[ -> one-to-one ;
coherence
cos | ].PI ,(2 * PI ).[ is one-to-one
cluster cos | ].0,(PI / 2).[ -> one-to-one ;
coherence
cos | ].0,(PI / 2).[ is one-to-one
cluster cos | ].(PI / 2),PI .[ -> one-to-one ;
coherence
cos | ].(PI / 2),PI .[ is one-to-one
cluster cos | ].PI ,((3 / 2) * PI ).[ -> one-to-one ;
coherence
cos | ].PI ,((3 / 2) * PI ).[ is one-to-one
cluster cos | ].((3 / 2) * PI ),(2 * PI ).[ -> one-to-one ;
coherence
cos | ].((3 / 2) * PI ),(2 * PI ).[ is one-to-one
end;
theorem Th61: :: SIN_COS6:61
:: deftheorem Def1 defines arcsin SIN_COS6:def 1 :
:: deftheorem Def2 defines arcsin SIN_COS6:def 2 :
theorem Th62: :: SIN_COS6:62
theorem Th63: :: SIN_COS6:63
theorem Th64: :: SIN_COS6:64
theorem Th65: :: SIN_COS6:65
theorem Th66: :: SIN_COS6:66
theorem Th67: :: SIN_COS6:67
theorem Th68: :: SIN_COS6:68
theorem Th69: :: SIN_COS6:69
theorem Th70: :: SIN_COS6:70
theorem Th71: :: SIN_COS6:71
theorem Th72: :: SIN_COS6:72
theorem Th73: :: SIN_COS6:73
theorem Th74: :: SIN_COS6:74
theorem Th75: :: SIN_COS6:75
theorem Th76: :: SIN_COS6:76
theorem Th77: :: SIN_COS6:77
theorem Th78: :: SIN_COS6:78
theorem Th79: :: SIN_COS6:79
theorem Th80: :: SIN_COS6:80
theorem Th81: :: SIN_COS6:81
theorem Th82: :: SIN_COS6:82
theorem Th83: :: SIN_COS6:83
theorem Th84: :: SIN_COS6:84
theorem Th85: :: SIN_COS6:85
:: deftheorem Def3 defines arccos SIN_COS6:def 3 :
:: deftheorem Def4 defines arccos SIN_COS6:def 4 :
theorem Th86: :: SIN_COS6:86
theorem Th87: :: SIN_COS6:87
theorem Th88: :: SIN_COS6:88
theorem Th89: :: SIN_COS6:89
theorem Th90: :: SIN_COS6:90
theorem Th91: :: SIN_COS6:91
theorem Th92: :: SIN_COS6:92
theorem Th93: :: SIN_COS6:93
theorem Th94: :: SIN_COS6:94
theorem Th95: :: SIN_COS6:95
theorem Th96: :: SIN_COS6:96
theorem Th97: :: SIN_COS6:97
theorem Th98: :: SIN_COS6:98
theorem Th99: :: SIN_COS6:99
theorem Th100: :: SIN_COS6:100
theorem Th101: :: SIN_COS6:101
theorem Th102: :: SIN_COS6:102
theorem Th103: :: SIN_COS6:103
theorem Th104: :: SIN_COS6:104
theorem Th105: :: SIN_COS6:105
theorem Th106: :: SIN_COS6:106
theorem Th107: :: SIN_COS6:107
theorem Th108: :: SIN_COS6:108
theorem Th109: :: SIN_COS6:109
theorem Th110: :: SIN_COS6:110
theorem Th111: :: SIN_COS6:111
theorem Th112: :: SIN_COS6:112