Article XCMPLX_1, MML version 4.99.1005
:: XCMPLX_1:th 1
theorem
for b1, b2, b3 being complex set holds
b1 + (b2 + b3) = (b1 + b2) + b3;
:: XCMPLX_1:th 2
theorem
for b1, b2, b3 being complex set
st b1 + b2 = b3 + b2
holds b1 = b3;
:: XCMPLX_1:th 3
theorem
for b1, b2 being complex set
st b1 = b1 + b2
holds b2 = 0;
:: XCMPLX_1:th 4
theorem
for b1, b2, b3 being complex set holds
b1 * (b2 * b3) = (b1 * b2) * b3;
:: XCMPLX_1:th 5
theorem
for b1, b2, b3 being complex set
st b1 <> 0 & b2 * b1 = b3 * b1
holds b2 = b3;
:: XCMPLX_1:th 6
theorem
for b1, b2 being complex set
st b1 * b2 = 0 & b1 <> 0
holds b2 = 0;
:: XCMPLX_1:th 7
theorem
for b1, b2 being complex set
st b1 <> 0 & b2 * b1 = b1
holds b2 = 1;
:: XCMPLX_1:th 8
theorem
for b1, b2, b3 being complex set holds
b1 * (b2 + b3) = (b1 * b2) + (b1 * b3);
:: XCMPLX_1:th 9
theorem
for b1, b2, b3, b4 being complex set holds
((b1 + b2) + b3) * b4 = ((b1 * b4) + (b2 * b4)) + (b3 * b4);
:: XCMPLX_1:th 10
theorem
for b1, b2, b3, b4 being complex set holds
(b1 + b2) * (b3 + b4) = (((b1 * b3) + (b1 * b4)) + (b2 * b3)) + (b2 * b4);
:: XCMPLX_1:th 11
theorem
for b1 being complex set holds
2 * b1 = b1 + b1;
:: XCMPLX_1:th 12
theorem
for b1 being complex set holds
3 * b1 = (b1 + b1) + b1;
:: XCMPLX_1:th 13
theorem
for b1 being complex set holds
4 * b1 = ((b1 + b1) + b1) + b1;
:: XCMPLX_1:th 14
theorem
for b1 being complex set holds
b1 - b1 = 0;
:: XCMPLX_1:th 15
theorem
for b1, b2 being complex set
st b1 - b2 = 0
holds b1 = b2;
:: XCMPLX_1:th 16
theorem
for b1, b2 being complex set
st b1 - b2 = b1
holds b2 = 0;
:: XCMPLX_1:th 17
theorem
for b1, b2 being complex set holds
b1 = b1 - (b2 - b2);
:: XCMPLX_1:th 18
theorem
for b1, b2 being complex set holds
b1 - (b1 - b2) = b2;
:: XCMPLX_1:th 19
theorem
for b1, b2, b3 being complex set
st b1 - b2 = b3 - b2
holds b1 = b3;
:: XCMPLX_1:th 20
theorem
for b1, b2, b3 being complex set
st b1 - b2 = b1 - b3
holds b2 = b3;
:: XCMPLX_1:th 21
theorem
for b1, b2, b3 being complex set holds
(b1 - b2) - b3 = (b1 - b3) - b2;
:: XCMPLX_1:th 22
theorem
for b1, b2, b3 being complex set holds
b1 - b2 = (b1 - b3) - (b2 - b3);
:: XCMPLX_1:th 23
theorem
for b1, b2, b3 being complex set holds
(b1 - b2) - (b1 - b3) = b3 - b2;
:: XCMPLX_1:th 24
theorem
for b1, b2, b3, b4 being complex set
st b1 - b2 = b3 - b4
holds b1 - b3 = b2 - b4;
:: XCMPLX_1:th 25
theorem
for b1, b2 being complex set holds
b1 = b1 + (b2 - b2);
:: XCMPLX_1:th 26
theorem
for b1, b2 being complex set holds
b1 = (b1 + b2) - b2;
:: XCMPLX_1:th 27
theorem
for b1, b2 being complex set holds
b1 = (b1 - b2) + b2;
:: XCMPLX_1:th 28
theorem
for b1, b2, b3 being complex set holds
b1 + b2 = (b1 + b3) + (b2 - b3);
:: XCMPLX_1:th 29
theorem
for b1, b2, b3 being complex set holds
(b1 + b2) - b3 = (b1 - b3) + b2;
:: XCMPLX_1:th 30
theorem
for b1, b2, b3 being complex set holds
(b1 - b2) + b3 = (b3 - b2) + b1;
:: XCMPLX_1:th 31
theorem
for b1, b2, b3 being complex set holds
b1 + b2 = (b1 + b3) - (b3 - b2);
:: XCMPLX_1:th 32
theorem
for b1, b2, b3 being complex set holds
b1 - b2 = (b1 + b3) - (b2 + b3);
:: XCMPLX_1:th 33
theorem
for b1, b2, b3, b4 being complex set
st b1 + b2 = b3 + b4
holds b1 - b3 = b4 - b2;
:: XCMPLX_1:th 34
theorem
for b1, b2, b3, b4 being complex set
st b1 - b2 = b3 - b4
holds b1 + b4 = b2 + b3;
:: XCMPLX_1:th 35
theorem
for b1, b2, b3, b4 being complex set
st b1 + b2 = b3 - b4
holds b1 + b4 = b3 - b2;
:: XCMPLX_1:th 36
theorem
for b1, b2, b3 being complex set holds
b1 - (b2 + b3) = (b1 - b2) - b3;
:: XCMPLX_1:th 37
theorem
for b1, b2, b3 being complex set holds
b1 - (b2 - b3) = (b1 - b2) + b3;
:: XCMPLX_1:th 38
theorem
for b1, b2, b3 being complex set holds
b1 - (b2 - b3) = b1 + (b3 - b2);
:: XCMPLX_1:th 39
theorem
for b1, b2, b3 being complex set holds
b1 - b2 = (b1 - b3) + (b3 - b2);
:: XCMPLX_1:th 40
theorem
for b1, b2, b3 being complex set holds
b1 * (b2 - b3) = (b1 * b2) - (b1 * b3);
:: XCMPLX_1:th 41
theorem
for b1, b2, b3, b4 being complex set holds
(b1 - b2) * (b3 - b4) = (b2 - b1) * (b4 - b3);
:: XCMPLX_1:th 42
theorem
for b1, b2, b3, b4 being complex set holds
((b1 - b2) - b3) * b4 = ((b1 * b4) - (b2 * b4)) - (b3 * b4);
:: XCMPLX_1:th 43
theorem
for b1, b2, b3, b4 being complex set holds
((b1 + b2) - b3) * b4 = ((b1 * b4) + (b2 * b4)) - (b3 * b4);
:: XCMPLX_1:th 44
theorem
for b1, b2, b3, b4 being complex set holds
((b1 - b2) + b3) * b4 = ((b1 * b4) - (b2 * b4)) + (b3 * b4);
:: XCMPLX_1:th 45
theorem
for b1, b2, b3, b4 being complex set holds
(b1 + b2) * (b3 - b4) = (((b1 * b3) - (b1 * b4)) + (b2 * b3)) - (b2 * b4);
:: XCMPLX_1:th 46
theorem
for b1, b2, b3, b4 being complex set holds
(b1 - b2) * (b3 + b4) = (((b1 * b3) + (b1 * b4)) - (b2 * b3)) - (b2 * b4);
:: XCMPLX_1:th 47
theorem
for b1, b2, b3, b4 being complex set holds
(b1 - b2) * (b3 - b4) = (((b1 * b3) - (b1 * b4)) - (b2 * b3)) + (b2 * b4);
:: XCMPLX_1:th 48
theorem
for b1, b2, b3 being complex set holds
(b1 / b2) / b3 = (b1 / b3) / b2;
:: XCMPLX_1:th 49
theorem
for b1 being complex set holds
b1 / 0 = 0;
:: XCMPLX_1:th 50
theorem
for b1, b2 being complex set
st b1 <> 0 & b2 <> 0
holds b1 / b2 <> 0;
:: XCMPLX_1:th 51
theorem
for b1, b2 being complex set
st b1 <> 0
holds b2 = b2 / (b1 / b1);
:: XCMPLX_1:th 52
theorem
for b1, b2 being complex set
st b1 <> 0
holds b1 / (b1 / b2) = b2;
:: XCMPLX_1:th 53
theorem
for b1, b2, b3 being complex set
st b1 <> 0 & b2 / b1 = b3 / b1
holds b2 = b3;
:: XCMPLX_1:th 54
theorem
for b1, b2 being complex set
st b1 / b2 <> 0
holds b2 = b1 / (b1 / b2);
:: XCMPLX_1:th 55
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 / b3 = (b2 / b1) / (b3 / b1);
:: XCMPLX_1:th 56
theorem
for b1 being complex set holds
1 / (1 / b1) = b1;
:: XCMPLX_1:th 57
theorem
for b1, b2 being complex set holds
1 / (b1 / b2) = b2 / b1;
:: XCMPLX_1:th 58
theorem
for b1, b2 being complex set
st b1 / b2 = 1
holds b1 = b2;
:: XCMPLX_1:th 59
theorem
for b1, b2 being complex set
st 1 / b1 = 1 / b2
holds b1 = b2;
:: XCMPLX_1:th 60
theorem
for b1 being complex set
st b1 <> 0
holds b1 / b1 = 1;
:: XCMPLX_1:th 61
theorem
for b1, b2 being complex set
st b1 <> 0 & b1 / b2 = b1
holds b2 = 1;
:: XCMPLX_1:th 62
theorem
for b1 being complex set holds
b1 / 0 = 0;
:: XCMPLX_1:th 63
theorem
for b1, b2, b3 being complex set holds
(b1 / b2) + (b3 / b2) = (b1 + b3) / b2;
:: XCMPLX_1:th 64
theorem
for b1, b2, b3, b4 being complex set holds
((b1 + b2) + b3) / b4 = ((b1 / b4) + (b2 / b4)) + (b3 / b4);
:: XCMPLX_1:th 65
theorem
for b1 being complex set holds
(b1 + b1) / 2 = b1;
:: XCMPLX_1:th 66
theorem
for b1 being complex set holds
(b1 / 2) + (b1 / 2) = b1;
:: XCMPLX_1:th 67
theorem
for b1, b2 being complex set
st b1 = (b1 + b2) / 2
holds b1 = b2;
:: XCMPLX_1:th 68
theorem
for b1 being complex set holds
((b1 + b1) + b1) / 3 = b1;
:: XCMPLX_1:th 69
theorem
for b1 being complex set holds
((b1 / 3) + (b1 / 3)) + (b1 / 3) = b1;
:: XCMPLX_1:th 70
theorem
for b1 being complex set holds
(((b1 + b1) + b1) + b1) / 4 = b1;
:: XCMPLX_1:th 71
theorem
for b1 being complex set holds
(((b1 / 4) + (b1 / 4)) + (b1 / 4)) + (b1 / 4) = b1;
:: XCMPLX_1:th 72
theorem
for b1 being complex set holds
(b1 / 4) + (b1 / 4) = b1 / 2;
:: XCMPLX_1:th 73
theorem
for b1 being complex set holds
(b1 + b1) / 4 = b1 / 2;
:: XCMPLX_1:th 74
theorem
for b1, b2 being complex set
st b1 * b2 = 1
holds b1 = 1 / b2;
:: XCMPLX_1:th 75
theorem
for b1, b2, b3 being complex set holds
b1 * (b2 / b3) = (b1 * b2) / b3;
:: XCMPLX_1:th 76
theorem
for b1, b2, b3 being complex set holds
(b1 / b2) * b3 = (b3 / b2) * b1;
:: XCMPLX_1:th 77
theorem
for b1, b2, b3, b4 being complex set holds
(b1 / b2) * (b3 / b4) = (b1 * b3) / (b2 * b4);
:: XCMPLX_1:th 78
theorem
for b1, b2, b3 being complex set holds
b1 / (b2 / b3) = (b1 * b3) / b2;
:: XCMPLX_1:th 79
theorem
for b1, b2, b3 being complex set holds
b1 / (b2 * b3) = (b1 / b2) / b3;
:: XCMPLX_1:th 80
theorem
for b1, b2, b3 being complex set holds
b1 / (b2 / b3) = b1 * (b3 / b2);
:: XCMPLX_1:th 81
theorem
for b1, b2, b3 being complex set holds
b1 / (b2 / b3) = (b3 / b2) * b1;
:: XCMPLX_1:th 82
theorem
for b1, b2, b3 being complex set holds
b1 / (b2 / b3) = b3 * (b1 / b2);
:: XCMPLX_1:th 83
theorem
for b1, b2, b3 being complex set holds
b1 / (b2 / b3) = (b1 / b2) * b3;
:: XCMPLX_1:th 84
theorem
for b1, b2, b3, b4 being complex set holds
(b1 * b2) / (b3 * b4) = ((b1 / b3) * b2) / b4;
:: XCMPLX_1:th 85
theorem
for b1, b2, b3, b4 being complex set holds
(b1 / b2) / (b3 / b4) = (b1 * b4) / (b2 * b3);
:: XCMPLX_1:th 86
theorem
for b1, b2, b3, b4 being complex set holds
(b1 / b2) * (b3 / b4) = (b1 / b4) * (b3 / b2);
:: XCMPLX_1:th 87
theorem
for b1, b2, b3, b4, b5 being complex set holds
b1 / ((b2 * b3) * (b4 / b5)) = (b5 / b3) * (b1 / (b2 * b4));
:: XCMPLX_1:th 88
theorem
for b1, b2 being complex set
st b1 <> 0
holds (b2 / b1) * b1 = b2;
:: XCMPLX_1:th 89
theorem
for b1, b2 being complex set
st b1 <> 0
holds b2 = b2 * (b1 / b1);
:: XCMPLX_1:th 90
theorem
for b1, b2 being complex set
st b1 <> 0
holds b2 = (b2 * b1) / b1;
:: XCMPLX_1:th 91
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 * b3 = (b2 * b1) * (b3 / b1);
:: XCMPLX_1:th 92
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 / b3 = (b2 * b1) / (b3 * b1);
:: XCMPLX_1:th 93
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 / b3 = (b2 / (b3 * b1)) * b1;
:: XCMPLX_1:th 94
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 * b3 = (b2 * b1) / (b1 / b3);
:: XCMPLX_1:th 95
theorem
for b1, b2, b3, b4 being complex set
st b1 <> 0 & b2 <> 0 & b3 * b1 = b4 * b2
holds b3 / b2 = b4 / b1;
:: XCMPLX_1:th 96
theorem
for b1, b2, b3, b4 being complex set
st b1 <> 0 & b2 <> 0 & b3 / b2 = b4 / b1
holds b3 * b1 = b4 * b2;
:: XCMPLX_1:th 97
theorem
for b1, b2, b3, b4 being complex set
st b1 <> 0 & b2 <> 0 & b3 * b1 = b4 / b2
holds b3 * b2 = b4 / b1;
:: XCMPLX_1:th 98
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 / b3 = b1 * ((b2 / b1) / b3);
:: XCMPLX_1:th 99
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 / b3 = (b2 / b1) * (b1 / b3);
:: XCMPLX_1:th 100
theorem
for b1, b2 being complex set holds
b1 * (1 / b2) = b1 / b2;
:: XCMPLX_1:th 101
theorem
for b1, b2 being complex set holds
b1 / (1 / b2) = b1 * b2;
:: XCMPLX_1:th 102
theorem
for b1, b2, b3 being complex set holds
(b1 / b2) * b3 = ((1 / b2) * b3) * b1;
:: XCMPLX_1:th 103
theorem
for b1, b2 being complex set holds
(1 / b1) * (1 / b2) = 1 / (b1 * b2);
:: XCMPLX_1:th 104
theorem
for b1, b2, b3 being complex set holds
(1 / b1) * (b2 / b3) = b2 / (b3 * b1);
:: XCMPLX_1:th 105
theorem
for b1, b2, b3 being complex set holds
(b1 / b2) / b3 = (1 / b2) * (b1 / b3);
:: XCMPLX_1:th 106
theorem
for b1, b2, b3 being complex set holds
(b1 / b2) / b3 = (1 / b3) * (b1 / b2);
:: XCMPLX_1:th 107
theorem
for b1 being complex set
st b1 <> 0
holds b1 * (1 / b1) = 1;
:: XCMPLX_1:th 108
theorem
for b1, b2 being complex set
st b1 <> 0
holds b2 = (b2 * b1) * (1 / b1);
:: XCMPLX_1:th 109
theorem
for b1, b2 being complex set
st b1 <> 0
holds b2 = b2 * ((1 / b1) * b1);
:: XCMPLX_1:th 110
theorem
for b1, b2 being complex set
st b1 <> 0
holds b2 = (b2 * (1 / b1)) * b1;
:: XCMPLX_1:th 111
theorem
for b1, b2 being complex set
st b1 <> 0
holds b2 = b2 / (b1 * (1 / b1));
:: XCMPLX_1:th 112
theorem
for b1, b2 being complex set
st b1 <> 0 & b2 <> 0
holds 1 / (b1 * b2) <> 0;
:: XCMPLX_1:th 113
theorem
for b1, b2 being complex set
st b1 <> 0 & b2 <> 0
holds (b1 / b2) * (b2 / b1) = 1;
:: XCMPLX_1:th 114
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds (b2 / b1) + b3 = (b2 + (b1 * b3)) / b1;
:: XCMPLX_1:th 115
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 + b3 = b1 * ((b2 / b1) + (b3 / b1));
:: XCMPLX_1:th 116
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 + b3 = ((b2 * b1) + (b3 * b1)) / b1;
:: XCMPLX_1:th 117
theorem
for b1, b2, b3, b4 being complex set
st b1 <> 0 & b2 <> 0
holds (b3 / b1) + (b4 / b2) = ((b3 * b2) + (b4 * b1)) / (b1 * b2);
:: XCMPLX_1:th 118
theorem
for b1, b2 being complex set
st b1 <> 0
holds b1 + b2 = b1 * (1 + (b2 / b1));
:: XCMPLX_1:th 119
theorem
for b1, b2 being complex set holds
(b1 / (2 * b2)) + (b1 / (2 * b2)) = b1 / b2;
:: XCMPLX_1:th 120
theorem
for b1, b2 being complex set holds
((b1 / (3 * b2)) + (b1 / (3 * b2))) + (b1 / (3 * b2)) = b1 / b2;
:: XCMPLX_1:th 121
theorem
for b1, b2, b3 being complex set holds
(b1 / b2) - (b3 / b2) = (b1 - b3) / b2;
:: XCMPLX_1:th 122
theorem
for b1 being complex set holds
b1 - (b1 / 2) = b1 / 2;
:: XCMPLX_1:th 123
theorem
for b1, b2, b3, b4 being complex set holds
((b1 - b2) - b3) / b4 = ((b1 / b4) - (b2 / b4)) - (b3 / b4);
:: XCMPLX_1:th 124
theorem
for b1, b2, b3, b4 being complex set
st b1 <> 0 & b2 <> 0 & b1 <> b2 & b3 / b1 = b4 / b2
holds b3 / b1 = (b3 - b4) / (b1 - b2);
:: XCMPLX_1:th 125
theorem
for b1, b2, b3, b4 being complex set holds
((b1 + b2) - b3) / b4 = ((b1 / b4) + (b2 / b4)) - (b3 / b4);
:: XCMPLX_1:th 126
theorem
for b1, b2, b3, b4 being complex set holds
((b1 - b2) + b3) / b4 = ((b1 / b4) - (b2 / b4)) + (b3 / b4);
:: XCMPLX_1:th 127
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds (b2 / b1) - b3 = (b2 - (b3 * b1)) / b1;
:: XCMPLX_1:th 128
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 - (b3 / b1) = ((b2 * b1) - b3) / b1;
:: XCMPLX_1:th 129
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 - b3 = b1 * ((b2 / b1) - (b3 / b1));
:: XCMPLX_1:th 130
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 - b3 = ((b2 * b1) - (b3 * b1)) / b1;
:: XCMPLX_1:th 131
theorem
for b1, b2, b3, b4 being complex set
st b1 <> 0 & b2 <> 0
holds (b3 / b1) - (b4 / b2) = ((b3 * b2) - (b4 * b1)) / (b1 * b2);
:: XCMPLX_1:th 132
theorem
for b1, b2 being complex set
st b1 <> 0
holds b1 - b2 = b1 * (1 - (b2 / b1));
:: XCMPLX_1:th 133
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds b2 = (((b1 * b2) + b3) - b3) / b1;
:: XCMPLX_1:th 134
theorem
for b1, b2 being complex set
st - b1 = - b2
holds b1 = b2;
:: XCMPLX_1:th 135
theorem
for b1 being complex set
st - b1 = 0
holds b1 = 0;
:: XCMPLX_1:th 136
theorem
for b1, b2 being complex set
st b1 + - b2 = 0
holds b1 = b2;
:: XCMPLX_1:th 137
theorem
for b1, b2 being complex set holds
b1 = (b1 + b2) + - b2;
:: XCMPLX_1:th 138
theorem
for b1, b2 being complex set holds
b1 = b1 + (b2 + - b2);
:: XCMPLX_1:th 139
theorem
for b1, b2 being complex set holds
b1 = ((- b2) + b1) + b2;
:: XCMPLX_1:th 140
theorem
for b1, b2 being complex set holds
- (b1 + b2) = (- b1) + - b2;
:: XCMPLX_1:th 141
theorem
for b1, b2 being complex set holds
- ((- b1) + b2) = b1 + - b2;
:: XCMPLX_1:th 142
theorem
for b1, b2 being complex set holds
b1 + b2 = - ((- b1) + - b2);
:: XCMPLX_1:th 143
theorem
for b1, b2 being complex set holds
- (b1 - b2) = b2 - b1;
:: XCMPLX_1:th 144
theorem
for b1, b2 being complex set holds
(- b1) - b2 = (- b2) - b1;
:: XCMPLX_1:th 145
theorem
for b1, b2 being complex set holds
b1 = (- b2) - ((- b1) - b2);
:: XCMPLX_1:th 146
theorem
for b1, b2, b3 being complex set holds
((- b1) - b2) - b3 = ((- b1) - b3) - b2;
:: XCMPLX_1:th 147
theorem
for b1, b2, b3 being complex set holds
((- b1) - b2) - b3 = ((- b2) - b3) - b1;
:: XCMPLX_1:th 148
theorem
for b1, b2, b3 being complex set holds
((- b1) - b2) - b3 = ((- b3) - b2) - b1;
:: XCMPLX_1:th 149
theorem
for b1, b2, b3 being complex set holds
(b1 - b2) - (b1 - b3) = - (b2 - b3);
:: XCMPLX_1:th 150
theorem
for b1 being complex set holds
0 - b1 = - b1;
:: XCMPLX_1:th 151
theorem
for b1, b2 being complex set holds
b1 + b2 = b1 - - b2;
:: XCMPLX_1:th 152
theorem
for b1, b2 being complex set holds
b1 = b1 - (b2 + - b2);
:: XCMPLX_1:th 153
theorem
for b1, b2, b3 being complex set
st b1 - b2 = b3 + - b2
holds b1 = b3;
:: XCMPLX_1:th 154
theorem
for b1, b2, b3 being complex set
st b1 - b2 = b1 + - b3
holds b2 = b3;
:: XCMPLX_1:th 155
theorem
for b1, b2, b3 being complex set holds
(b1 + b2) - b3 = ((- b3) + b1) + b2;
:: XCMPLX_1:th 156
theorem
for b1, b2, b3 being complex set holds
(b1 - b2) + b3 = ((- b2) + b3) + b1;
:: XCMPLX_1:th 157
theorem
for b1, b2, b3 being complex set holds
b1 - ((- b2) - b3) = (b1 + b2) + b3;
:: XCMPLX_1:th 158
theorem
for b1, b2, b3 being complex set holds
(b1 - b2) - b3 = ((- b2) - b3) + b1;
:: XCMPLX_1:th 159
theorem
for b1, b2, b3 being complex set holds
(b1 - b2) - b3 = ((- b3) + b1) - b2;
:: XCMPLX_1:th 160
theorem
for b1, b2, b3 being complex set holds
(b1 - b2) - b3 = ((- b3) - b2) + b1;
:: XCMPLX_1:th 161
theorem
for b1, b2 being complex set holds
- (b1 + b2) = (- b2) - b1;
:: XCMPLX_1:th 162
theorem
for b1, b2 being complex set holds
- (b1 - b2) = (- b1) + b2;
:: XCMPLX_1:th 163
theorem
for b1, b2 being complex set holds
- ((- b1) + b2) = b1 - b2;
:: XCMPLX_1:th 164
theorem
for b1, b2 being complex set holds
b1 + b2 = - ((- b1) - b2);
:: XCMPLX_1:th 165
theorem
for b1, b2, b3 being complex set holds
((- b1) + b2) - b3 = ((- b3) + b2) - b1;
:: XCMPLX_1:th 166
theorem
for b1, b2, b3 being complex set holds
((- b1) + b2) - b3 = ((- b3) - b1) + b2;
:: XCMPLX_1:th 167
theorem
for b1, b2, b3 being complex set holds
- ((b1 + b2) + b3) = ((- b1) - b2) - b3;
:: XCMPLX_1:th 168
theorem
for b1, b2, b3 being complex set holds
- ((b1 + b2) - b3) = ((- b1) - b2) + b3;
:: XCMPLX_1:th 169
theorem
for b1, b2, b3 being complex set holds
- ((b1 - b2) + b3) = ((- b1) + b2) - b3;
:: XCMPLX_1:th 170
theorem
for b1, b2, b3 being complex set holds
- ((b1 - b2) - b3) = ((- b1) + b2) + b3;
:: XCMPLX_1:th 171
theorem
for b1, b2, b3 being complex set holds
- (((- b1) + b2) + b3) = (b1 - b2) - b3;
:: XCMPLX_1:th 172
theorem
for b1, b2, b3 being complex set holds
- (((- b1) + b2) - b3) = (b1 - b2) + b3;
:: XCMPLX_1:th 173
theorem
for b1, b2, b3 being complex set holds
- (((- b1) - b2) + b3) = (b1 + b2) - b3;
:: XCMPLX_1:th 174
theorem
for b1, b2, b3 being complex set holds
- (((- b1) - b2) - b3) = (b1 + b2) + b3;
:: XCMPLX_1:th 175
theorem
for b1, b2 being complex set holds
(- b1) * b2 = - (b1 * b2);
:: XCMPLX_1:th 176
theorem
for b1, b2 being complex set holds
(- b1) * b2 = b1 * - b2;
:: XCMPLX_1:th 177
theorem
for b1, b2 being complex set holds
(- b1) * - b2 = b1 * b2;
:: XCMPLX_1:th 178
theorem
for b1, b2 being complex set holds
- (b1 * - b2) = b1 * b2;
:: XCMPLX_1:th 179
theorem
for b1, b2 being complex set holds
- ((- b1) * b2) = b1 * b2;
:: XCMPLX_1:th 180
theorem
for b1 being complex set holds
(- 1) * b1 = - b1;
:: XCMPLX_1:th 181
theorem
for b1 being complex set holds
(- b1) * - 1 = b1;
:: XCMPLX_1:th 182
theorem
for b1, b2 being complex set
st b1 <> 0 & b2 * b1 = - b1
holds b2 = - 1;
:: XCMPLX_1:th 183
theorem
for b1 being complex set
st b1 * b1 = 1 & b1 <> 1
holds b1 = - 1;
:: XCMPLX_1:th 184
theorem
for b1 being complex set holds
(- b1) + (2 * b1) = b1;
:: XCMPLX_1:th 185
theorem
for b1, b2, b3 being complex set holds
(b1 - b2) * b3 = (b2 - b1) * - b3;
:: XCMPLX_1:th 186
theorem
for b1, b2, b3 being complex set holds
(b1 - b2) * b3 = - ((b2 - b1) * b3);
:: XCMPLX_1:th 187
theorem
for b1 being complex set holds
b1 - (2 * b1) = - b1;
:: XCMPLX_1:th 188
theorem
for b1, b2 being complex set holds
- (b1 / b2) = (- b1) / b2;
:: XCMPLX_1:th 189
theorem
for b1, b2 being complex set holds
b1 / - b2 = - (b1 / b2);
:: XCMPLX_1:th 190
theorem
for b1, b2 being complex set holds
- (b1 / - b2) = b1 / b2;
:: XCMPLX_1:th 191
theorem
for b1, b2 being complex set holds
- ((- b1) / b2) = b1 / b2;
:: XCMPLX_1:th 192
theorem
for b1, b2 being complex set holds
(- b1) / - b2 = b1 / b2;
:: XCMPLX_1:th 193
theorem
for b1, b2 being complex set holds
(- b1) / b2 = b1 / - b2;
:: XCMPLX_1:th 194
theorem
for b1 being complex set holds
- b1 = b1 / - 1;
:: XCMPLX_1:th 195
theorem
for b1 being complex set holds
b1 = (- b1) / - 1;
:: XCMPLX_1:th 196
theorem
for b1, b2 being complex set
st b1 / b2 = - 1
holds b1 = - b2 & b2 = - b1;
:: XCMPLX_1:th 197
theorem
for b1, b2 being complex set
st b1 <> 0 & b1 / b2 = - b1
holds b2 = - 1;
:: XCMPLX_1:th 198
theorem
for b1 being complex set
st b1 <> 0
holds (- b1) / b1 = - 1;
:: XCMPLX_1:th 199
theorem
for b1 being complex set
st b1 <> 0
holds b1 / - b1 = - 1;
:: XCMPLX_1:th 200
theorem
for b1 being complex set
st b1 <> 0 & b1 = 1 / b1 & b1 <> 1
holds b1 = - 1;
:: XCMPLX_1:th 201
theorem
for b1, b2, b3, b4 being complex set
st b1 <> 0 & b2 <> 0 & b1 <> - b2 & b3 / b1 = b4 / b2
holds b3 / b1 = (b3 + b4) / (b1 + b2);
:: XCMPLX_1:th 202
theorem
for b1, b2 being complex set
st b1 " = b2 "
holds b1 = b2;
:: XCMPLX_1:th 203
theorem
0 " = 0;
:: XCMPLX_1:th 204
theorem
for b1, b2 being complex set
st b1 <> 0
holds b2 = (b2 * b1) * (b1 ");
:: XCMPLX_1:th 205
theorem
for b1, b2 being complex set holds
b1 " * (b2 ") = (b1 * b2) ";
:: XCMPLX_1:th 206
theorem
for b1, b2 being complex set holds
(b1 * (b2 ")) " = b1 " * b2;
:: XCMPLX_1:th 207
theorem
for b1, b2 being complex set holds
(b1 " * (b2 ")) " = b1 * b2;
:: XCMPLX_1:th 208
theorem
for b1, b2 being complex set
st b1 <> 0 & b2 <> 0
holds b1 * (b2 ") <> 0;
:: XCMPLX_1:th 209
theorem
for b1, b2 being complex set
st b1 <> 0 & b2 <> 0
holds b1 " * (b2 ") <> 0;
:: XCMPLX_1:th 210
theorem
for b1, b2 being complex set
st b1 * (b2 ") = 1
holds b1 = b2;
:: XCMPLX_1:th 211
theorem
for b1, b2 being complex set
st b1 * b2 = 1
holds b1 = b2 ";
:: XCMPLX_1:th 213
theorem
for b1, b2 being complex set
st b1 <> 0 & b2 <> 0
holds b1 " + (b2 ") = (b1 + b2) * ((b1 * b2) ");
:: XCMPLX_1:th 214
theorem
for b1, b2 being complex set
st b1 <> 0 & b2 <> 0
holds b1 " - (b2 ") = (b2 - b1) * ((b1 * b2) ");
:: XCMPLX_1:th 215
theorem
for b1, b2 being complex set holds
(b1 / b2) " = b2 / b1;
:: XCMPLX_1:th 216
theorem
for b1, b2 being complex set holds
b1 " / (b2 ") = b2 / b1;
:: XCMPLX_1:th 217
theorem
for b1 being complex set holds
1 / b1 = b1 ";
:: XCMPLX_1:th 218
theorem
for b1 being complex set holds
1 / (b1 ") = b1;
:: XCMPLX_1:th 219
theorem
for b1 being complex set holds
(1 / b1) " = b1;
:: XCMPLX_1:th 220
theorem
for b1, b2 being complex set
st 1 / b1 = b2 "
holds b1 = b2;
:: XCMPLX_1:th 221
theorem
for b1, b2 being complex set holds
b1 / (b2 ") = b1 * b2;
:: XCMPLX_1:th 222
theorem
for b1, b2, b3 being complex set holds
b1 " * (b2 / b3) = b2 / (b1 * b3);
:: XCMPLX_1:th 223
theorem
for b1, b2 being complex set holds
b1 " / b2 = (b1 * b2) ";
:: XCMPLX_1:th 224
theorem
for b1 being complex set holds
(- b1) " = - (b1 ");
:: XCMPLX_1:th 225
theorem
for b1 being complex set
st b1 <> 0 & b1 = b1 " & b1 <> 1
holds b1 = - 1;
:: XCMPLX_1:th 226
theorem
for b1, b2, b3 being complex set holds
((b1 + b2) + b3) - b2 = b1 + b3;
:: XCMPLX_1:th 227
theorem
for b1, b2, b3 being complex set holds
((b1 - b2) + b3) + b2 = b1 + b3;
:: XCMPLX_1:th 228
theorem
for b1, b2, b3 being complex set holds
((b1 + b2) - b3) - b2 = b1 - b3;
:: XCMPLX_1:th 229
theorem
for b1, b2, b3 being complex set holds
((b1 - b2) - b3) + b2 = b1 - b3;
:: XCMPLX_1:th 230
theorem
for b1, b2 being complex set holds
(b1 - b1) - b2 = - b2;
:: XCMPLX_1:th 231
theorem
for b1, b2 being complex set holds
((- b1) + b1) - b2 = - b2;
:: XCMPLX_1:th 232
theorem
for b1, b2 being complex set holds
(b1 - b2) - b1 = - b2;
:: XCMPLX_1:th 233
theorem
for b1, b2 being complex set holds
((- b1) - b2) + b1 = - b2;
:: XCMPLX_1:th 234
theorem
for b1, b2 being complex set
st b2 <> 0
holds ex b3 being complex set st
b1 = b2 / b3;
:: XCMPLX_1:th 235
theorem
for b1, b2, b3, b4, b5 being complex set holds
b1 / ((b2 * b3) * (b4 / b5)) = (b5 / b3) * (b1 / (b2 * b4));
:: XCMPLX_1:th 236
theorem
for b1, b2, b3, b4 being complex set holds
(((b1 - b2) / b3) * b4) + b2 = ((1 - (b4 / b3)) * b2) + ((b4 / b3) * b1);
:: XCMPLX_1:th 237
theorem
for b1, b2, b3 being complex set
st b1 <> 0
holds (b1 * b2) + b3 = b1 * (b2 + (b3 / b1));