:: Complex Numbers - Basic Theorems
::  by Library Committee
::
:: Received April 10, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users
::           (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.

environ

 vocabularies XCMPLX_0, ARYTM_3, CARD_1, RELAT_1, ARYTM_1;
 notations ORDINAL1, NUMBERS, XCMPLX_0;
 constructors NUMBERS, XCMPLX_0, XXREAL_0, XREAL_0;
 registrations XCMPLX_0, XXREAL_0, XREAL_0, ORDINAL1;
 requirements SUBSET, NUMERALS, ARITHM;


begin

reserve a, b, c, d, e for Complex;

:: '+' operation only

theorem :: XCMPLX_1:1 :: AXIOMS'13
  a + (b + c) = (a + b) + c;

theorem :: XCMPLX_1:2 :: REAL_1'10
  a + c = b + c implies a = b;

theorem :: XCMPLX_1:3 :: INT_1'24
  a = a + b implies b = 0;

:: using operation '*'

theorem :: XCMPLX_1:4
  a * (b * c) = (a * b) * c;

theorem :: XCMPLX_1:5 :: REAL_1'9
  c <> 0 & a * c = b * c implies a = b;

theorem :: XCMPLX_1:6 :: REAL_1'23  :: right to left - requirements REAL
  a*b=0 implies a=0 or b=0;

theorem :: XCMPLX_1:7  :: REAL_2'37
  b <> 0 & a * b = b implies a = 1;

:: operations '+' and '*' only

theorem :: XCMPLX_1:8 :: AXIOMS'18
  a * (b + c) = a * b + a * c;

theorem :: XCMPLX_1:9 :: REAL_2'99_1
  (a + b + c) * d = a * d + b * d + c * d;

theorem :: XCMPLX_1:10 :: REAL_2'101_1
  (a + b) * (c + d) = a * c + a * d + b * c + b * d;

theorem :: XCMPLX_1:11 :: SQUARE_1'5
  2 * a = a + a;

theorem :: XCMPLX_1:12 :: REAL_2'88_1
  3 * a = a + a + a;

theorem :: XCMPLX_1:13 :: REAL_2'88_2
  4 * a = a + a + a + a;

:: using operation '-'

theorem :: XCMPLX_1:14 :: REAL_1'36
  a - a = 0;

theorem :: XCMPLX_1:15 :: SQUARE_1'8
  a - b = 0 implies a = b;

theorem :: XCMPLX_1:16 :: REAL_2'1
  b - a = b implies a = 0;

:: 2 times '-'

theorem :: XCMPLX_1:17 :: REAL_2'17_2
  a = a - (b - b);

theorem :: XCMPLX_1:18 :: SEQ_4'3
  a - (a - b) = b;

theorem :: XCMPLX_1:19 :: REAL_2'2_3
  a - c = b - c implies a = b;

theorem :: XCMPLX_1:20 :: REAL_2'2_5
  c - a = c - b implies a = b;

theorem :: XCMPLX_1:21 :: REAL_2'24_1
  a - b - c = a - c - b;

theorem :: XCMPLX_1:22 :: REAL_2'29_1
  a - c = (a - b) - (c - b);

theorem :: XCMPLX_1:23 :: JGRAPH_6'1_1
  (c - a) - (c - b) = b - a;

theorem :: XCMPLX_1:24 :: REAL_2'15
  a - b = c - d implies a - c = b - d;

:: using operations '-' and '+'

theorem :: XCMPLX_1:25 :: REAL_2'17_1
  a = a + (b - b);

theorem :: XCMPLX_1:26 :: REAL_1'30
  a = a + b - b;

theorem :: XCMPLX_1:27 :: SQUARE_1'6
  a = a - b + b;

theorem :: XCMPLX_1:28 :: REAL_2'28_1
  a + c = a + b + (c - b);

:: 2 times '-'

theorem :: XCMPLX_1:29 :: REAL_2'22_1, INT_1'1, REAL_1'17
  a + b - c = a - c + b;

theorem :: XCMPLX_1:30 :: REAL_2'23_1
  a - b + c = c - b + a;

theorem :: XCMPLX_1:31 :: REAL_2'28_2
  a + c = a + b - (b - c);

theorem :: XCMPLX_1:32 :: REAL_2'29_3
  a - c = a + b - (c + b);

theorem :: XCMPLX_1:33 :: REAL_2'13
  a + b = c + d implies a - c = d - b;

theorem :: XCMPLX_1:34 :: REAL_2'14
  a - c = d - b implies a + b = c + d;

theorem :: XCMPLX_1:35 :: REAL_2'16
  a + b = c - d implies a + d = c - b;

:: 3 times '-'

theorem :: XCMPLX_1:36 :: REAL_1'27
  a - (b + c) = a - b - c;

theorem :: XCMPLX_1:37 :: REAL_1'28
  a - (b - c) = a - b + c;

theorem :: XCMPLX_1:38 :: REAL_2'18
  a - (b - c) = a + (c - b);

theorem :: XCMPLX_1:39 :: REAL_2'29_2
  a - c = (a - b) + (b - c);

theorem :: XCMPLX_1:40 :: REAL_1'29
  a * (b - c) = a * b - a * c;

theorem :: XCMPLX_1:41 :: REAL_2'98
  (a - b) * (c - d) = (b - a) * (d - c);

theorem :: XCMPLX_1:42 :: REAL_2'99_4
  (a - b - c) * d = a * d - b * d - c * d;

:: using operations '-' and '*', '+'

theorem :: XCMPLX_1:43 :: REAL_2'99_2
  (a + b - c) * d = a * d + b * d - c * d;

theorem :: XCMPLX_1:44 :: REAL_2'99_3
  (a - b + c) * d = a * d - b * d + c * d;

theorem :: XCMPLX_1:45 :: REAL_2'101_2
  (a + b) * (c - d) = a * c - a * d + b * c - b * d;

theorem :: XCMPLX_1:46 :: REAL_2'101_3
  (a - b) * (c + d) = a * c + a * d - b * c - b * d;

theorem :: XCMPLX_1:47 :: REAL_2'101_4
  (a - b) * (e - d) = a * e - a * d - b * e + b * d;

:: using operation '/'

theorem :: XCMPLX_1:48 :: REAL_2'67_1
  a / b / c = a / c / b;

:: 0

theorem :: XCMPLX_1:49  :: REAL_2'19
  a / 0 = 0;

theorem :: XCMPLX_1:50 :: REAL_2'42_2
  a <> 0 & b <> 0 implies a / b <> 0;

:: 2 times '/'

theorem :: XCMPLX_1:51 :: REAL_2'62_4
  b <> 0 implies a = a / (b / b);

theorem :: XCMPLX_1:52 :: TOPREAL6'5
  a <> 0 implies a / (a / b) = b;

theorem :: XCMPLX_1:53 :: REAL_2'31
  c <> 0 & a / c = b / c implies a = b;

theorem :: XCMPLX_1:54 :: REAL_2'74
  a / b <> 0 implies b = a / (a / b);

theorem :: XCMPLX_1:55 :: REAL_2'55_1
  c <> 0 implies a / b = (a / c) / (b / c);

:: 1

theorem :: XCMPLX_1:56 :: SQUARE_1'16
  1 / (1 / a) = a;

theorem :: XCMPLX_1:57 :: REAL_2'48_1
  1 / (a / b) = b / a;

theorem :: XCMPLX_1:58
  a / b = 1 implies a = b;

theorem :: XCMPLX_1:59  :: REAL_2'33_2
  1 / a = 1 / b implies a = b;

:: 0 and 1

theorem :: XCMPLX_1:60 :: REAL_1'37
  a <> 0 implies a / a = 1;

theorem :: XCMPLX_1:61 :: REAL_2'39
  b <> 0 & b / a = b implies a = 1;

:: using operations '/' and '+'

theorem :: XCMPLX_1:62  :: REAL_1'40_1
  a / c + b / c = (a + b) / c;

theorem :: XCMPLX_1:63 :: REAL_2'100_1
  (a + b + e) / d = a / d + b / d + e / d;

:: 2

theorem :: XCMPLX_1:64 :: SQUARE_1'15
  (a + a) / 2 = a;

theorem :: XCMPLX_1:65 :: SEQ_2'2_1
  a/2 + a/2 = a;

theorem :: XCMPLX_1:66 :: TOPREAL3'4
  a = (a + b) / 2 implies a = b;

:: 3

theorem :: XCMPLX_1:67 :: REAL_2'89_1
  (a + a + a)/3 = a;

theorem :: XCMPLX_1:68 :: SEQ_4'5
  a/3 + a/3 + a/3 = a;

:: 4

theorem :: XCMPLX_1:69 :: REAL_2'89_2
  (a + a + a + a) / 4 = a;

theorem :: XCMPLX_1:70 :: REAL_2'90
  a/4 + a/4 + a/4 + a/4 = a;

theorem :: XCMPLX_1:71 :: SEQ_2'2_2
  a / 4 + a / 4 = a / 2;

theorem :: XCMPLX_1:72 :: REAL_2'89_3
  (a + a) / 4 = a / 2;

:: using operations '/' and '*'

theorem :: XCMPLX_1:73 :: REAL_2'35_1
  a * b = 1 implies a = 1 / b;

theorem :: XCMPLX_1:74 :: SQUARE_1'18
  a * (b / c) = (a * b) / c;

theorem :: XCMPLX_1:75 :: REAL_2'80_1
  a / b * e = e / b * a;

:: 3 times '/'

theorem :: XCMPLX_1:76 :: REAL_1'35
  (a / b) * (c / d) = (a * c) / (b * d);

theorem :: XCMPLX_1:77 :: REAL_1'42
  a / (b / c) = (a * c) / b;

theorem :: XCMPLX_1:78  :: SQUARE_1'17
  a / (b * c) = a / b / c;

theorem :: XCMPLX_1:79 :: REAL_2'61_1
  a / (b / c) = a * (c / b);

theorem :: XCMPLX_1:80 :: REAL_2'61_2
  a / (b / c) = c / b * a;

theorem :: XCMPLX_1:81  :: REAL_2'61_3
  a / (b / e) = e * (a / b);

theorem :: XCMPLX_1:82 :: REAL_2'61_4
  a / (b / c) = a / b * c;

theorem :: XCMPLX_1:83 :: REAL_2'70
  (a * b) / (c * d) = (a / c * b) / d;

:: 4 times '/'

theorem :: XCMPLX_1:84 :: REAL_1'82
  (a / b) / (c / d) = (a * d) / (b * c);

theorem :: XCMPLX_1:85 :: REAL_2'53
  (a / c) * (b / d) = (a / d) * (b / c);

theorem :: XCMPLX_1:86 :: IRRAT_1'5
  a / (b * c * (d / e)) = (e / c) * (a / (b * d));

:: 0

theorem :: XCMPLX_1:87 :: REAL_1'43
  b <> 0 implies a / b * b = a;

theorem :: XCMPLX_1:88 :: REAL_2'62_1
  b <> 0 implies a = a * (b / b);

theorem :: XCMPLX_1:89 :: REAL_2'62_2
  b <> 0 implies a = a * b / b;

theorem :: XCMPLX_1:90 :: REAL_2'78
  b <> 0 implies a * c = a * b * (c / b);

:: 2 times '/'

theorem :: XCMPLX_1:91 :: REAL_1'38
  c <> 0 implies a / b = (a * c) / (b * c);

theorem :: XCMPLX_1:92 :: REAL_2'55_2
  c <> 0 implies a / b = a / (b * c) * c;

theorem :: XCMPLX_1:93 :: REAL_2'79
  b <> 0 implies a * c = a * b / (b / c);

theorem :: XCMPLX_1:94  :: REAL_2'75
  c <> 0 & d <> 0 & a * c = b * d implies a / d = b / c;

theorem :: XCMPLX_1:95  :: REAL_2'76
  c <> 0 & d<>0 & a/d=b/c implies a*c = b*d;

theorem :: XCMPLX_1:96 :: REAL_2'77
  c <> 0 & d <> 0 & a * c = b / d implies a * d = b / c;

:: 3 times '/'

theorem :: XCMPLX_1:97 :: REAL_2'55_3
  c <> 0 implies a / b = c * (a / c / b);

theorem :: XCMPLX_1:98 :: REAL_2'55
  c <> 0 implies a / b = a / c * (c / b);

:: 1

theorem :: XCMPLX_1:99 :: REAL_2'56:
  a * (1 / b) = a / b;

theorem :: XCMPLX_1:100 :: REAL_2'57
  a / (1 / b) = a * b;

theorem :: XCMPLX_1:101 :: REAL_2'80_3
  a / b * c = 1 / b * c * a;

:: 3 times '/'

theorem :: XCMPLX_1:102 :: REAL_2'51
  (1 / a) * (1 / b) = 1 / (a * b);

theorem :: XCMPLX_1:103 :: REAL_2'67_4
  1 / c * (a / b) = a / (b * c);

:: 4 times '/'

theorem :: XCMPLX_1:104 :: REAL_2'67_2
  a / b / c = 1 / b * (a / c);

theorem :: XCMPLX_1:105 :: REAL_2'67_3
  a / b / c = 1 / c * (a / b);

:: 1 and 0

theorem :: XCMPLX_1:106  :: REAL_1'34
  a <> 0 implies a * (1 / a) = 1;

theorem :: XCMPLX_1:107 :: REAL_2'62_3
  b <> 0 implies a = a * b * (1 / b);

theorem :: XCMPLX_1:108 :: REAL_2'62_6
  b <> 0 implies a = a * (1 / b * b);

theorem :: XCMPLX_1:109 :: REAL_2'62_7
  b <> 0 implies a = a * (1 / b) * b;

theorem :: XCMPLX_1:110 :: REAL_2'62_5
  b <> 0 implies a = a / (b * (1 / b));

theorem :: XCMPLX_1:111 :: REAL_2'42_4
  a <> 0 & b <> 0 implies 1 / (a * b) <> 0;

theorem :: XCMPLX_1:112 :: JGRAPH_2'1
  a <> 0 & b <> 0 implies (a / b) * (b / a) = 1;

:: using operations '*', '+' and '/'

theorem :: XCMPLX_1:113  :: REAL_2'65
  b <> 0 implies a / b + c = (a + b * c) / b;

theorem :: XCMPLX_1:114  :: REAL_2'92
  c <> 0 implies a + b = c * (a / c + b / c);

theorem :: XCMPLX_1:115  :: REAL_2'94
  c <> 0 implies a + b = (a * c + b * c) / c;

theorem :: XCMPLX_1:116  :: REAL_1'41_1
  b <> 0 & d <> 0 implies a / b + c / d =(a * d + c * b) / (b * d );

theorem :: XCMPLX_1:117  :: REAL_2'96
  a <> 0 implies a + b = a * (1 + b / a);

:: 2

theorem :: XCMPLX_1:118 :: REAL_2'91_1
  a / (2 * b) + a / (2 * b) = a / b;

:: 3

theorem :: XCMPLX_1:119 :: REAL_2'91_2
  a / (3 * b) + a / (3 * b) + a / (3 * b) = a / b;

theorem :: XCMPLX_1:120  :: REAL_1'40_2
  a / c - b / c = (a - b) / c;

theorem :: XCMPLX_1:121 :: TOPREAL6'4
  a - a / 2 = a / 2;

theorem :: XCMPLX_1:122 :: REAL_2'100_4
  (a - b - c) / d = a / d - b / d - c / d;

theorem :: XCMPLX_1:123 :: REAL_2'82
  b <> 0 & d <> 0 & b <> d & a / b = e / d implies a / b = (a - e) / (b - d );

:: using operations '-', '/' and '+'

theorem :: XCMPLX_1:124 :: REAL_2'100_2
  (a + b - e) / d = a / d + b / d - e / d;

theorem :: XCMPLX_1:125 :: REAL_2'100_3
  (a - b + e) / d = a / d - b / d + e / d;

:: using operations '-', '/' and '*'

theorem :: XCMPLX_1:126  :: REAL_2'66_1
  b <> 0 implies a / b - e = (a - e * b) / b;

theorem :: XCMPLX_1:127 :: REAL_2'66_2
  b <> 0 implies c - a / b = (c * b - a) / b;

theorem :: XCMPLX_1:128 :: REAL_2'93
  c <> 0 implies a - b = c * (a / c - b / c);

theorem :: XCMPLX_1:129 :: REAL_2'95
  c <> 0 implies a - b = (a * c - b * c) / c;

theorem :: XCMPLX_1:130 :: REAL_1'41_2
  b <> 0 & d <> 0 implies a / b - c / d = (a * d - c * b) / (b * d);

theorem :: XCMPLX_1:131 :: REAL_2'97
  a <> 0 implies a - b = a * (1 - b / a);

:: using operation '-', '/', '*' and '+'

theorem :: XCMPLX_1:132 :: POLYEQ_1'24
  a <> 0 implies c = (a * c + b - b) / a;

:: using unary operation '-'

theorem :: XCMPLX_1:133 :: REAL_2'2_2
  -a = -b implies a = b;

theorem :: XCMPLX_1:134 :: REAL_1'22:  :: right to left - requirements REAL
  -a = 0 implies a = 0;

theorem :: XCMPLX_1:135 :: REAL_2'2_1
  a + -b = 0 implies a = b;

theorem :: XCMPLX_1:136 :: REAL_2'11
  a = a + b + -b;

theorem :: XCMPLX_1:137 :: REAL_2'17_1
  a = a + (b + -b);

theorem :: XCMPLX_1:138 :: INT_1'3
  a = (- b + a) + b;

theorem :: XCMPLX_1:139 :: REAL_2'6_1
  - (a + b) = -a + -b;

theorem :: XCMPLX_1:140 :: REAL_2'9_2
  - (-a + b) = a + -b;

theorem :: XCMPLX_1:141 :: REAL_2'10_2
  a+b=-(-a+-b);

:: using unary and binary operation '-'

theorem :: XCMPLX_1:142 :: REAL_1'83
  -(a - b) = b - a;

theorem :: XCMPLX_1:143 :: REAL_2'5
  - a - b = - b - a;

theorem :: XCMPLX_1:144 :: REAL_2'17_4
  a = - b - (- a - b);

:: binary '-' 4 times

theorem :: XCMPLX_1:145 :: REAL_2'26_1
  - a - b - c = - a - c - b;

theorem :: XCMPLX_1:146 :: REAL_2'26_2
  - a - b - c = - b - c - a;

theorem :: XCMPLX_1:147 :: REAL_2'26_4
  - a - b - c = - c - b - a;

theorem :: XCMPLX_1:148 :: JGRAPH_6'1_2
  (c - a) - (c - b) = - (a - b);

:: 0

theorem :: XCMPLX_1:149 :: REAL_1'19
  0 - a = - a;

:: using unary and binary operations '-' and '+'

theorem :: XCMPLX_1:150 :: REAL_2'10_3
  a + b = a - - b;

theorem :: XCMPLX_1:151 :: REAL_2'17_3
  a = a - (b + -b);

theorem :: XCMPLX_1:152 :: REAL_2'2_4
  a - c = b + - c implies a = b;

theorem :: XCMPLX_1:153 :: REAL_2'2_6
  c - a = c + - b implies a = b;

:: '+' 3 times

theorem :: XCMPLX_1:154 :: REAL_2'22_2
  a + b - c = - c + a + b;

theorem :: XCMPLX_1:155 :: REAL_2'23_2
  a - b + c = - b + c + a;

theorem :: XCMPLX_1:156 :: REAL_2'20_2
  a - (- b - c) = a + b + c;

:: binary '-' 3 times

theorem :: XCMPLX_1:157 :: REAL_2'20_1
  a - b - c = - b - c + a;

theorem :: XCMPLX_1:158 :: REAL_2'24_3
  a - b - c = - c + a - b;

theorem :: XCMPLX_1:159 :: REAL_2'24_4
  a - b - c = - c - b + a;

:: using unary and binary operations '-' and '+'

theorem :: XCMPLX_1:160 :: REAL_2'6_2
  - (a + b) = - b - a;

theorem :: XCMPLX_1:161 :: REAL_2'8
  - (a - b) = - a + b;

theorem :: XCMPLX_1:162 :: REAL_2'9_1
  -(-a+b)=a-b;

theorem :: XCMPLX_1:163 :: REAL_2'10_1
  a + b = -(- a - b);

theorem :: XCMPLX_1:164 :: REAL_2'25_1
  - a + b - c = - c + b - a;

:: using unary and binary operations '-' and '+' (both '-' 2 times)

theorem :: XCMPLX_1:165 :: REAL_2'25_2
  - a + b - c = - c - a + b;

theorem :: XCMPLX_1:166 :: REAL_2'27_1
  - (a + b + c) = - a - b - c;

theorem :: XCMPLX_1:167 :: REAL_2'27_2
  - (a + b - c) = - a - b + c;

theorem :: XCMPLX_1:168 :: REAL_2'27_3
  - (a - b + c) = - a + b - c;

theorem :: XCMPLX_1:169 :: REAL_2'27_5
  - (a - b - c) = - a + b + c;

theorem :: XCMPLX_1:170 :: REAL_2'27_4
  - (- a + b + c) = a - b - c;

theorem :: XCMPLX_1:171 :: REAL_2'27_6
  - (- a + b - c) = a - b + c;

theorem :: XCMPLX_1:172 :: REAL_2'27_7
  - (- a - b + c) = a + b - c;

theorem :: XCMPLX_1:173 :: REAL_2'27_8
  - (- a - b - c) = a + b + c;

:: using unary operations '-' and '*'

theorem :: XCMPLX_1:174 :: REAL_1'21_1
  (- a) * b = -(a * b);

theorem :: XCMPLX_1:175 :: REAL_1'21_2
  (- a) * b = a * (- b);

theorem :: XCMPLX_1:176 :: REAL_2'49_1
  (- a) * (- b) = a * b;

theorem :: XCMPLX_1:177 :: REAL_2'49_2
  - a * (- b) = a * b;

theorem :: XCMPLX_1:178 :: REAL_2'49_3
  -(-a) * b = a * b;

theorem :: XCMPLX_1:179 :: REAL_2'71_1
  (-1) * a = -a;

theorem :: XCMPLX_1:180 :: REAL_2'71_2
  (- a) * (- 1) = a;

theorem :: XCMPLX_1:181  :: REAL_2'38
  b<>0 & a*b=-b implies a=-1;

:: Thx

theorem :: XCMPLX_1:182
  a * a = 1 implies a = 1 or a = -1;

theorem :: XCMPLX_1:183 :: TOPREAL6'3
  -a + 2 * a = a;

theorem :: XCMPLX_1:184 :: REAL_2'85_1
  (a - b) * c = (b - a) * (- c);

theorem :: XCMPLX_1:185 :: REAL_2'85_2
  (a - b) * c = - (b - a) * c;

theorem :: XCMPLX_1:186 :: TOPREAL6'2
  a - 2 * a = -a;

:: using unary operations '-' and '/'

theorem :: XCMPLX_1:187 :: REAL_1'39_1
  -a / b = (-a) / b;

theorem :: XCMPLX_1:188  :: REAL_1'39_2
  a / (- b) = -a / b;

theorem :: XCMPLX_1:189 :: REAL_2'58_1
  - a / (- b) = a / b;

theorem :: XCMPLX_1:190  :: REAL_2'58_2
  -(- a) / b = a / b;

theorem :: XCMPLX_1:191 :: REAL_2'58_3
  (- a) / (- b) = a / b;

theorem :: XCMPLX_1:192 :: REAL_2'58
  (-a) / b = a / (-b);

theorem :: XCMPLX_1:193 :: REAL_2'71_3
  -a = a / (-1);

theorem :: XCMPLX_1:194 :: REAL_2'71
  a = (- a) / (-1);

theorem :: XCMPLX_1:195 :: REAL_2'34
  a / b = - 1 implies a = - b & b = - a;

theorem :: XCMPLX_1:196 :: REAL_2'40
  b <> 0 & b / a = - b implies a = -1;

theorem :: XCMPLX_1:197 :: REAL_2'45_2
  a <> 0 implies (-a) / a = -1;

theorem :: XCMPLX_1:198 :: REAL_2'45_3
  a <> 0 implies a / (- a) = -1;

theorem :: XCMPLX_1:199 :: REAL_2'46_2
  a <> 0 & a = 1 / a implies a = 1 or a = -1;

theorem :: XCMPLX_1:200 :: REAL_2'83:
  b <> 0 & d <> 0 & b <> -d & a / b = e / d implies a / b = (a + e) / (b + d );

:: using operation '"'

theorem :: XCMPLX_1:201 :: REAL_2'33_1
  a" = b" implies a = b;

theorem :: XCMPLX_1:202 :: REAL_1'31
  0" = 0;

:: using '"' and '*'

theorem :: XCMPLX_1:203
  b <> 0 implies a = a*b*b";

theorem :: XCMPLX_1:204 :: REAL_1'24
  a" * b" = (a * b)";

theorem :: XCMPLX_1:205 :: REAL_2'47_1
  (a * b")" = a" * b;

theorem :: XCMPLX_1:206 :: REAL_2'47_2
  (a" * b")" = a * b;

theorem :: XCMPLX_1:207 :: REAL_2'42_1
  a <> 0 & b <> 0 implies a * b" <> 0;

theorem :: XCMPLX_1:208 :: REAL_2'42_3
  a <> 0 & b <> 0 implies a" * b" <> 0;

theorem :: XCMPLX_1:209 :: REAL_2'30_2
  a * b" = 1 implies a = b;

theorem :: XCMPLX_1:210 :: REAL_2'35_2
  a * b = 1 implies a = b";

:: using '"', '*', and '+'

theorem :: XCMPLX_1:211
  a <> 0 & b <> 0 implies a" + b" = (a + b)*(a*b)";

:: using '"', '*', and '-'

theorem :: XCMPLX_1:212
  a <> 0 & b <> 0 implies a" - b" = (b - a)*(a*b)";

:: using '"' and '/'

theorem :: XCMPLX_1:213 :: REAL_1'81
  (a / b)" = b / a;

theorem :: XCMPLX_1:214
  (a"/b") = b/a;

theorem :: XCMPLX_1:215 :: REAL_1'33_1
  1 / a = a";

theorem :: XCMPLX_1:216 :: REAL_1'33_2
  1 / a" = a;

theorem :: XCMPLX_1:217 :: REAL_2'36_21
  (1 / a)" = a;

theorem :: XCMPLX_1:218 :: REAL_2'33_3
  1 / a = b" implies a = b;

:: using '"', '*', and '/'

theorem :: XCMPLX_1:219
  a/b" = a*b;

theorem :: XCMPLX_1:220
  a"*(c/b) = c/(a*b);

theorem :: XCMPLX_1:221
  a"/b = (a*b)";

:: both unary operations

theorem :: XCMPLX_1:222 :: REAL_2'45_1
  (- a)" = -a";

theorem :: XCMPLX_1:223 :: REAL_2'46_1
  a <> 0 & a = a" implies a = 1 or a = -1;

begin :: additional
:: from JORDAN4

theorem :: XCMPLX_1:224
  a+b+c-b=a+c;

theorem :: XCMPLX_1:225
  a-b+c+b=a+c;

theorem :: XCMPLX_1:226
  a+b-c-b=a-c;

theorem :: XCMPLX_1:227
  a-b-c+b=a-c;

theorem :: XCMPLX_1:228
  a-a-b=-b;

theorem :: XCMPLX_1:229
  -a+a-b=-b;

theorem :: XCMPLX_1:230
  a-b-a=-b;

theorem :: XCMPLX_1:231
  -a-b+a=-b;

begin :: Addenda
:: from REAL_2, 2005.02.05, A.T.

theorem :: XCMPLX_1:232
  for a,b st b<>0 ex e st a=b/e;

:: from IRRAT_1, 2005.02.05, A.T.

theorem :: XCMPLX_1:233
  a/(b*c*(d/e))=(e/c)*(a/(b*d));

:: from BORSUK_6, 2005.02.05, A.T.

theorem :: XCMPLX_1:234
  ((d - c)/b) * a + c = (1 - a/b)*c + (a/b) * d;

:: Missing, 2005.07.04, A.T.

theorem :: XCMPLX_1:235
  a <> 0 implies a * b + c = a * (b + (c/a));