### Introduction to Arithmetics

by
Andrzej Trybulec

Copyright (c) 2003 Association of Mizar Users

MML identifier: ARYTM_0
[ MML identifier index ]

```environ

vocabulary ARYTM_0, COMPLEX1, FUNCT_2, FUNCT_1, FUNCOP_1, ARYTM_2, BOOLE,
ARYTM_1, ARYTM_3, ORDINAL2, ORDINAL1, OPPCAT_1, RELAT_1;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2,
FUNCT_4, ORDINAL1, ARYTM_3, ARYTM_2, ARYTM_1, NUMBERS;
constructors ARYTM_1, FRAENKEL, FUNCT_4, XBOOLE_0, NUMBERS;
clusters XBOOLE_0, ARYTM_2, SUBSET_1, ORDINAL2, ARYTM_3, FRAENKEL, FUNCT_2,
FUNCT_4, ZFMISC_1, NUMBERS;
requirements BOOLE, SUBSET;
theorems XBOOLE_0, ARYTM_1, ZFMISC_1, TARSKI, ARYTM_2, ORDINAL2, XBOOLE_1,
ORDINAL1, ORDINAL3, ARYTM_3, FUNCT_2, FUNCT_4, FUNCT_1, ENUMSET1,
NUMBERS;

begin :: Arithmetics

Lm1: {} in {{}} by TARSKI:def 1;
Lm2:
now let e be set;
assume
A1: one = [0,e];
{0} = 0 \/ {0}
.= succ 0 by ORDINAL1:def 1
.= {{0},{0,e}} by A1,ORDINAL2:def 4,TARSKI:def 5;
then {0} in {0} by TARSKI:def 2;
end;

theorem Th1:
REAL+ c= REAL
proof
A1: REAL+ c= REAL+ \/ [:{{}},REAL+:] by XBOOLE_1:7;
not [{},{}] in REAL+ by ARYTM_2:3;
hence thesis by A1,NUMBERS:def 1,ZFMISC_1:40;
end;

theorem Th2:
for x being Element of REAL+ st x <> {} holds [{},x] in REAL
proof let x be Element of REAL+ such that
A1: x <> {};
A2: now assume [{},x] in {[{},{}]};
then [{},x] = [{},{}] by TARSKI:def 1;
end;
{} in {{}} by TARSKI:def 1;
then [{},x] in [:{{}},REAL+:] by ZFMISC_1:106;
then [{},x] in REAL+ \/ [:{{}},REAL+:] by XBOOLE_0:def 2;
hence [{},x] in REAL by A2,NUMBERS:def 1,XBOOLE_0:def 4;
end;

theorem Th3:
for y being set st [{},y] in REAL holds y <> {}
proof let y be set such that
A1: [{},y] in REAL and
A2: y = {};
[{},y] in {[{},{}]} by A2,TARSKI:def 1;
hence contradiction by A1,NUMBERS:def 1,XBOOLE_0:def 4;
end;

theorem Th4:
for x,y being Element of REAL+ holds x - y in REAL
proof let x,y be Element of REAL+;
per cases;
suppose y <=' x;
then x - y = x -' y by ARYTM_1:def 2;
then x - y in REAL+;
hence x - y in REAL by Th1;
suppose
A1: not y <=' x;
then A2: x - y = [{},y -' x] by ARYTM_1:def 2;
y -' x <> {} by A1,ARYTM_1:9;
hence x - y in REAL by A2,Th2;
end;

theorem Th5:
REAL+ misses [:{{}},REAL+:]
proof
assume REAL+ meets [:{{}},REAL+:];
then consider x being set such that
A1: x in REAL+ and
A2: x in [:{{}},REAL+:] by XBOOLE_0:3;
consider x1,x2 being set such that
A3: x1 in {{}} and
A4: x2 in REAL+ & x = [x1,x2] by A2,ZFMISC_1:103;
x1 = {} by A3,TARSKI:def 1;
end;

begin  :: Real numbers

theorem Th6:
for x,y being Element of REAL+ st x - y = {} holds x = y
proof let x,y be Element of REAL+;
A1: {} <> [{},y -' x];
assume
A2: x - y = {};
then A3:  y <=' x by A1,ARYTM_1:def 2;
then x -' y = {} by A2,ARYTM_1:def 2;
hence x = y by A3,ARYTM_1:10;
end;

theorem Th7:
not ex a,b being set st one = [a,b]
proof let a,b be set;
assume one = [a,b];
then A1:  {{}} = {{a,b},{a}} by ORDINAL3:18,TARSKI:def 5;
{a} in {{a,b},{a}} by TARSKI:def 2;
end;

theorem Th8:
for x,y,z being Element of REAL+ st x <> {} & x *' y = x *' z
holds y = z
proof let x,y,z be Element of REAL+;
assume that
A1: x <> {} and
A2: x *' y = x *' z;
per cases;
suppose
A3: z <=' y;
then x *' (y -' z) = (x *' y) - (x *' z) by ARYTM_1:26
.= {} by A2,ARYTM_1:18;
then {} = y -' z by A1,ARYTM_1:2
.= y - z by A3,ARYTM_1:def 2;
hence y = z by Th6;
suppose
A4: y <=' z;
then x *' (z -' y) = x *' z - x *' y by ARYTM_1:26
.= {} by A2,ARYTM_1:18;
then {} = z -' y by A1,ARYTM_1:2
.= z - y by A4,ARYTM_1:def 2;
hence y = z by Th6;
end;

canceled;

begin :: from XREAL_0

definition let x,y be Element of REAL;
canceled;
func +(x,y) -> Element of REAL means :Def2:
ex x',y' being Element of REAL+ st x = x' & y = y' & it = x' + y'
if x in REAL+ & y in REAL+,
ex x',y' being Element of REAL+ st x = x' & y = [0,y'] & it = x' - y'
if x in REAL+ & y in [:{0},REAL+:],
ex x',y' being Element of REAL+ st x = [0,x'] & y = y' & it = y' - x'
if y in REAL+ & x in [:{0},REAL+:]
otherwise
ex x',y' being Element of REAL+
st x = [0,x'] & y = [0,y'] & it = [0,x'+y'];
existence
proof
hereby assume x in REAL+ & y in REAL+;
then reconsider x'=x, y'=y as Element of REAL+;
x' + y' in REAL+;
then reconsider IT = x' + y' as Element of REAL by Th1;
take IT,x',y';
thus x = x' & y = y' & IT = x' + y';
end;
hereby assume x in REAL+;
then reconsider x'=x as Element of REAL+;
assume y in [:{0},REAL+:];
then consider z,y' being set such that
A1:    z in{0} and
A2:    y' in REAL+ and
A3:    y = [z,y'] by ZFMISC_1:103;
reconsider y' as Element of REAL+ by A2;
reconsider IT = x' - y' as Element of REAL by Th4;
take IT,x',y';
thus x = x' & y = [0,y'] & IT = x' - y' by A1,A3,TARSKI:def 1;
end;
hereby assume y in REAL+;
then reconsider y'=y as Element of REAL+;
assume x in [:{0},REAL+:];
then consider z,x' being set such that
A4:    z in{0} and
A5:    x' in REAL+ and
A6:    x = [z,x'] by ZFMISC_1:103;
reconsider x' as Element of REAL+ by A5;
reconsider IT = y' - x' as Element of REAL by Th4;
take IT,x',y';
thus x = [0,x'] & y = y' & IT = y' - x' by A4,A6,TARSKI:def 1;
end;
x in REAL+ \/ [:{0},REAL+:] & y in REAL+ \/ [:{0},REAL+:]
by NUMBERS:def 1,XBOOLE_0:def 4;
then A7:  (x in REAL+ or x in [:{0},REAL+:]) &
(y in REAL+ or y in [:{0},REAL+:]) by XBOOLE_0:def 2;
assume
A8:   not(x in REAL+ & y in REAL+) &
not(x in REAL+ & y in [:{0},REAL+:]) &
not(y in REAL+ & x in [:{0},REAL+:]);
then consider z1,x' being set such that
A9:    z1 in{0} and
A10:    x' in REAL+ and
A11:    x = [z1,x'] by A7,ZFMISC_1:103;
reconsider x' as Element of REAL+ by A10;
consider z2,y' being set such that
A12:    z2 in{0} and
A13:    y' in REAL+ and
A14:    y = [z2,y'] by A7,A8,ZFMISC_1:103;
reconsider y' as Element of REAL+ by A13;
x = [0,x'] by A9,A11,TARSKI:def 1;
then x' <> 0 by Th3;
then x' + y' <> 0 by ARYTM_2:6;
then reconsider IT = [0,y' + x'] as Element of REAL by Th2;
take IT,x',y';
thus x = [0,x'] & y = [0,y'] & IT = [0,x' + y']
by A9,A11,A12,A14,TARSKI:def 1;
end;
uniqueness
proof let IT1,IT2 be Element of REAL;
thus x in REAL+ & y in REAL+ &
(ex x',y' being Element of REAL+ st x = x' & y = y' & IT1 = x' + y') &
(ex x',y' being Element of REAL+ st x = x' & y = y' & IT2 = x' + y')
implies IT1 = IT2;
thus x in REAL+ & y in [:{0},REAL+:] &
(ex x',y' being Element of REAL+ st x = x' & y = [0,y'] &IT1 = x' - y') &
(ex x'',y'' being Element of REAL+ st
x = x'' & y = [0,y''] & IT2 = x'' - y'')
implies IT1 = IT2 by ZFMISC_1:33;
thus y in REAL+ & x in [:{0},REAL+:] &
(ex x',y' being Element of REAL+ st x = [0,x'] & y = y' & IT1 = y' - x') &
(ex x'',y'' being Element of REAL+ st
x = [0,x''] & y = y'' & IT2 = y'' - x'')
implies IT1 = IT2 by ZFMISC_1:33;
assume
not(x in REAL+ & y in REAL+) &
not(x in REAL+ & y in [:{0},REAL+:]) &
not(y in REAL+ & x in [:{0},REAL+:]);
given x',y' being Element of REAL+ such that
A15:  x = [0,x'] & y = [0,y'] and
A16:  IT1 = [0,x'+y'];
given x'',y'' being Element of REAL+ such that
A17:  x = [0,x''] & y = [0,y''] and
A18:  IT2 = [0,x''+y''];
x' = x'' & y' = y'' by A15,A17,ZFMISC_1:33;
hence thesis by A16,A18;
end;
consistency by Th5,XBOOLE_0:3;
commutativity;

func *(x,y) -> Element of REAL means :Def3:
ex x',y' being Element of REAL+ st x = x' & y = y' & it = x' *' y'
if x in REAL+ & y in REAL+,
ex x',y' being Element of REAL+ st x = x' & y = [0,y'] & it = [0,x' *' y']
if x in REAL+ & y in [:{0},REAL+:] & x <> 0,
ex x',y' being Element of REAL+ st x = [0,x'] & y = y' & it = [0,y' *' x']
if y in REAL+ & x in [:{0},REAL+:] & y <> 0,
ex x',y' being Element of REAL+ st x = [0,x'] & y = [0,y'] & it = y' *' x'
if x in [:{0},REAL+:] & y in [:{0},REAL+:]
otherwise it = 0;
existence
proof
hereby assume x in REAL+ & y in REAL+;
then reconsider x'=x, y'=y as Element of REAL+;
x' *' y' in REAL+;
then reconsider IT = x' *' y' as Element of REAL by Th1;
take IT,x',y';
thus x = x' & y = y' & IT = x' *' y';
end;
hereby assume x in REAL+;
then reconsider x'=x as Element of REAL+;
assume y in [:{0},REAL+:];
then consider z,y' being set such that
A19:    z in{0} and
A20:    y' in REAL+ and
A21:    y = [z,y'] by ZFMISC_1:103;
reconsider y' as Element of REAL+ by A20;
assume
A22:    x <> 0;
y = [0,y'] by A19,A21,TARSKI:def 1;
then y' <> 0 by Th3;
then x' *' y' <> 0 by A22,ARYTM_1:2;
then reconsider IT = [0,x' *' y'] as Element of REAL by Th2;
take IT,x',y';
thus x = x' & y = [0,y'] & IT = [0,x' *' y'] by A19,A21,TARSKI:def 1;
end;
hereby assume y in REAL+;
then reconsider y'=y as Element of REAL+;
assume x in [:{0},REAL+:];
then consider z,x' being set such that
A23:    z in{0} and
A24:    x' in REAL+ and
A25:    x = [z,x'] by ZFMISC_1:103;
reconsider x' as Element of REAL+ by A24;
assume
A26:    y <> 0;
x = [0,x'] by A23,A25,TARSKI:def 1;
then x' <> 0 by Th3;
then x' *' y' <> 0 by A26,ARYTM_1:2;
then reconsider IT = [0,y' *' x'] as Element of REAL by Th2;
take IT,x',y';
thus x = [0,x'] & y = y' & IT = [0,y' *' x'] by A23,A25,TARSKI:def 1;
end;
hereby assume x in [:{0},REAL+:];
then consider z1,x' being set such that
A27:    z1 in{0} and
A28:    x' in REAL+ and
A29:    x = [z1,x'] by ZFMISC_1:103;
reconsider x' as Element of REAL+ by A28;
assume y in [:{0},REAL+:];
then consider z2,y' being set such that
A30:    z2 in{0} and
A31:    y' in REAL+ and
A32:    y = [z2,y'] by ZFMISC_1:103;
reconsider y' as Element of REAL+ by A31;
x' *' y' in REAL+;
then reconsider IT = y' *' x' as Element of REAL by Th1;
take IT,x',y';
thus x = [0,x'] & y = [0,y'] & IT = y' *' x'
by A27,A29,A30,A32,TARSKI:def 1;
end;
thus thesis by Th1,ARYTM_2:21;
end;
uniqueness
proof let IT1,IT2 be Element of REAL;
thus x in REAL+ & y in REAL+ &
(ex x',y' being Element of REAL+ st x = x' & y = y' & IT1 = x' *' y') &
(ex x',y' being Element of REAL+ st x = x' & y = y' & IT2 = x' *' y')
implies IT1 = IT2;
thus x in REAL+ & y in [:{0},REAL+:] & x <> 0 &
(ex x',y' being Element of REAL+
st x = x' & y = [0,y'] & IT1 = [0,x' *' y']) &
(ex x'',y'' being Element of REAL+
st x = x'' & y = [0,y''] & IT2 = [0,x'' *' y''])
implies IT1 = IT2 by ZFMISC_1:33;
thus y in REAL+ & x in [:{0},REAL+:] & y <> 0 &
(ex x',y' being Element of REAL+
st x = [0,x'] & y = y' & IT1 = [0,y' *' x']) &
(ex x'',y'' being Element of REAL+
st x = [0,x''] & y = y'' & IT2 = [0,y'' *' x''])
implies IT1 = IT2 by ZFMISC_1:33;
hereby assume
x in [:{0},REAL+:] & y in [:{0},REAL+:];
given x',y' being Element of REAL+ such that
A33:  x = [0,x'] & y = [0,y'] and
A34:  IT1 = y' *' x';
given x'',y'' being Element of REAL+ such that
A35:  x = [0,x''] & y = [0,y''] and
A36:  IT2 = y'' *' x'';
x' = x'' & y' = y'' by A33,A35,ZFMISC_1:33;
hence IT1 = IT2 by A34,A36;
end;
thus thesis;
end;
consistency by Th5,XBOOLE_0:3;
commutativity;
end;

reserve x,y for Element of REAL;

definition let x be Element of REAL;
func opp x -> Element of REAL means
:Def4: +(x,it) = 0;
existence
proof
reconsider z' = 0 as Element of REAL+ by ARYTM_2:21;
A1:   x in REAL+ \/ [:{0},REAL+:] & not x in {[0,0]}
by NUMBERS:def 1,XBOOLE_0:def 4;
per cases by A1,XBOOLE_0:def 2;
suppose
A2:   x = 0;
then reconsider x' = x as Element of REAL+ by ARYTM_2:21;
take x;
x' + x' = 0 by A2,ARYTM_2:def 8;
hence thesis by Def2,Th1,ARYTM_2:21;
suppose that
A3:   x in REAL+ and
A4:  x <> 0;
0 in {0} by TARSKI:def 1;
then A5:   [0,x] in [:{0},REAL+:] by A3,ZFMISC_1:106;
then A6:   [0,x] in REAL+ \/ [:{0},REAL+:] by XBOOLE_0:def 2;
[0,x] <> [0,0] by A4,ZFMISC_1:33;
then reconsider y = [0,x] as Element of REAL
by A6,NUMBERS:def 1,ZFMISC_1:64;
take y;
reconsider x' = x as Element of REAL+ by A3;
A7:   x' <=' x';
z' + x' = x' by ARYTM_2:def 8;
then z' = x' -' x' by A7,ARYTM_1:def 1;
then 0 = x' - x' by A7,ARYTM_1:def 2;
hence thesis by A5,Def2,Th1,ARYTM_2:21;
suppose
A8:   x in [:{0},REAL+:];
then consider a,b being set such that
A9:   a in {0} and
A10:   b in REAL+ and
A11:   x = [a,b] by ZFMISC_1:103;
reconsider y = b as Element of REAL by A10,Th1;
take y;
now per cases;
case x in REAL+ & y in REAL+;
hence ex x',y' being Element of REAL+ st x = x' & y = y' & 0 = x' + y'
by A8,Th5,XBOOLE_0:3;
case x in REAL+ & y in [:{0},REAL+:];
hence ex x',y' being Element of REAL+ st x = x' & y = [0,y'] &
0 = x' - y' by A10,Th5,XBOOLE_0:3;
case y in REAL+ & x in [:{0},REAL+:];
reconsider x' = b, y' = y as Element of REAL+ by A10;
take x', y';
thus x = [0,x'] by A9,A11,TARSKI:def 1;
thus y = y';
thus 0 = y' - x' by ARYTM_1:18;
case not(x in REAL+ & y in REAL+) &
not(x in REAL+ & y in [:{0},REAL+:]) &
not (y in REAL+ & x in [:{0},REAL+:]);
hence ex x',y' being Element of REAL+
st x = [0,x'] & y = [0,y'] & 0 = [0,y'+x'] by A8,A10;
end;
hence thesis by Def2,Th1,ARYTM_2:21;
end;
uniqueness
proof let y,z be Element of REAL such that
A12: +(x,y) = 0 and
A13: +(x,z) = 0;
per cases;
suppose x in REAL+ & y in REAL+ & z in REAL+;
then (ex x',y' being Element of REAL+ st x = x' & y = y' & 0 = x' + y')
& ex x',y' being Element of REAL+ st x = x' & z = y' & 0 = x' + y'
by A12,A13,Def2;
hence y = z by ARYTM_2:12;
suppose that
A14:  x in REAL+ and
A15:  y in REAL+ and
A16:  z in [:{0},REAL+:];
consider x',y' being Element of REAL+ such that
A17:   x = x' and y = y' and
A18:   0 = x' + y' by A12,A14,A15,Def2;
consider x'',y'' being Element of REAL+ such that
A19:   x = x'' and
A20:   z = [0,y''] and
A21:   0 = x'' - y'' by A13,A14,A16,Def2;
A22:   x'' = 0 by A17,A18,A19,ARYTM_2:6;
[0,0] in {[0,0]} by TARSKI:def 1;
then A23:    not [0,0] in REAL by NUMBERS:def 1,XBOOLE_0:def 4;
z in REAL;
hence y = z by A20,A21,A22,A23,ARYTM_1:19;
suppose that
A24:  x in REAL+ and
A25:  z in REAL+ and
A26:  y in [:{0},REAL+:];
consider x',z' being Element of REAL+ such that
A27:   x = x' and z = z' and
A28:   0 = x' + z' by A13,A24,A25,Def2;
consider x'',y' being Element of REAL+ such that
A29:   x = x'' and
A30:   y = [0,y'] and
A31:   0 = x'' - y' by A12,A24,A26,Def2;
A32:   x'' = 0 by A27,A28,A29,ARYTM_2:6;
[0,0] in {[0,0]} by TARSKI:def 1;
then A33:    not [0,0] in REAL by NUMBERS:def 1,XBOOLE_0:def 4;
y in REAL;
hence z = y by A30,A31,A32,A33,ARYTM_1:19;
suppose that
A34:  x in REAL+ and
A35:  y in [:{0},REAL+:] and
A36:  z in [:{0},REAL+:];
consider x',y' being Element of REAL+ such that
A37:   x = x' and
A38:   y = [0,y'] and
A39:   0 = x' - y' by A12,A34,A35,Def2;
consider x'',z' being Element of REAL+ such that
A40:   x = x'' and
A41:   z = [0,z'] and
A42:   0 = x'' - z' by A13,A34,A36,Def2;
y' = x' by A39,Th6 .= z' by A37,A40,A42,Th6;
hence y = z by A38,A41;
suppose that
A43:  z in REAL+ and
A44:  y in REAL+ and
A45:  x in [:{0},REAL+:];
consider x',y' being Element of REAL+ such that
A46:   x = [0,x'] and
A47:   y = y' and
A48:   0 = y' - x' by A12,A44,A45,Def2;
consider x'',z' being Element of REAL+ such that
A49:   x = [0,x''] and
A50:   z = z' and
A51:   0 = z' - x'' by A13,A43,A45,Def2;
x' = x'' by A46,A49,ZFMISC_1:33;
then z' = x' by A51,Th6 .= y' by A48,Th6;
hence y = z by A47,A50;
suppose not(x in REAL+ & y in REAL+) &
not(x in REAL+ & y in [:{0},REAL+:]) &
not(y in REAL+ & x in [:{0},REAL+:]);
then ex x',y' being Element of REAL+
st x = [0,x'] & y = [0,y'] & 0 = [0,x'+y'] by A12,Def2;
hence y = z;
suppose not(x in REAL+ & z in REAL+) &
not(x in REAL+ & z in [:{0},REAL+:]) &
not(z in REAL+ & x in [:{0},REAL+:]);
then ex x',z' being Element of REAL+
st x = [0,x'] & z = [0,z'] & 0 = [0,x'+z'] by A13,Def2;
hence y = z;
end;
involutiveness;
func inv x -> Element of REAL means
:Def5: *(x,it) = one if x <> 0
otherwise it = 0;
existence
proof
thus x <> 0 implies ex y st *(x,y) = one
proof assume
A52:   x <> 0;
A53:   x in REAL+ \/ [:{0},REAL+:] by NUMBERS:def 1,XBOOLE_0:def 4;
per cases by A53,XBOOLE_0:def 2;
suppose
A54:  x in REAL+;
then reconsider x' = x as Element of REAL+;
consider y' being Element of REAL+ such that
A55:  x' *' y' = one by A52,ARYTM_2:15;
y' in REAL+;
then reconsider y = y' as Element of REAL by Th1;
take y;
consider x'',y'' being Element of REAL+ such that
A56: x = x'' and
A57: y = y'' and
A58: *(x,y) = x'' *' y'' by A54,Def3;
thus *(x,y) = one by A55,A56,A57,A58;
suppose
A59: x in [:{0},REAL+:];
then consider x1,x2 being set such that
x1 in {0} and
A60: x2 in REAL+ and
A61: x = [x1,x2] by ZFMISC_1:103;
reconsider x2 as Element of REAL+ by A60;
not x in {[0,0]} by NUMBERS:def 1,XBOOLE_0:def 4;
then A62: x <> [0,0] by TARSKI:def 1;
x1 in {0} by A59,A61,ZFMISC_1:106;
then x2 <> 0 by A61,A62,TARSKI:def 1;
then consider y2 being Element of REAL+ such that
A63:  x2 *' y2 = one by ARYTM_2:15;
A64: [0,y2] in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then A65: [0,y2] in REAL+ \/ [:{0},REAL+:] by XBOOLE_0:def 2;
now assume [0,y2] in {[0,0]};
then [0,y2] = [0,0] by TARSKI:def 1;
then y2 = 0 by ZFMISC_1:33;
end;
then reconsider y = [0,y2] as Element of REAL
by A65,NUMBERS:def 1,XBOOLE_0:def 4;
consider x',y' being Element of REAL+ such that
A66: x = [0,x'] and
A67: y = [0,y'] and
A68: *(x,y) = y' *' x' by A59,A64,Def3;
take y;
y' = y2 & x' = x2 by A61,A66,A67,ZFMISC_1:33;
hence *(x,y) = one by A63,A68;
end;
thus thesis;
end;
uniqueness
proof let y,z be Element of REAL;
thus x <> 0 & *(x,y) = one & *(x,z) = one implies y = z
proof assume that
A69:    x <> 0 and
A70:    *(x,y) = one and
A71:    *(x,z) = one;
A72:     for x,y being Element of REAL st *(x,y) =one holds x <> 0
proof let x,y be Element of REAL such that
A73:       *(x,y) =one and
A74:       x = 0;
A75:       not x in [:{0},REAL+:] by A74,Th5,ARYTM_2:21,XBOOLE_0:3;
A76:      y in REAL+ \/ [:{0},REAL+:] by NUMBERS:def 1,XBOOLE_0:def 4;
per cases by A76,XBOOLE_0:def 2;
suppose y in REAL+;
then ex x',y' being Element of REAL+
st x = x' & y = y' & one = x' *' y' by A73,A74,Def3,ARYTM_2:21;
suppose y in [:{0},REAL+:];
then not y in REAL+ by Th5,XBOOLE_0:3;
end;
then A77:    y <> 0 by A70;
A78:    z <> 0 by A71,A72;
per cases;
suppose x in REAL+ & y in REAL+ & z in REAL+;
then (ex x',y' being Element of REAL+ st x = x' & y = y' & one = x' *'
y')
& ex x',y' being Element of REAL+ st x = x' & z = y' & one = x' *' y'
by A70,A71,Def3;
hence y = z by A69,Th8;
suppose that
A79:    x in [:{0},REAL+:] and
A80:    y in [:{0},REAL+:] and
A81:    z in [:{0},REAL+:];
consider x',y' being Element of REAL+ such that
A82:     x = [0,x'] and
A83:     y = [0,y'] and
A84:     one = y' *' x' by A70,A79,A80,Def3;
consider x'',z' being Element of REAL+ such that
A85:     x = [0,x''] and
A86:     z = [0,z'] and
A87:     one = z' *' x'' by A71,A79,A81,Def3;
A88:     x' = x'' by A82,A85,ZFMISC_1:33;
x' <> 0 by A82,Th3;
hence y = z by A83,A84,A86,A87,A88,Th8;
suppose x in REAL+ & y in [:{0},REAL+:];
then ex x',y' being Element of REAL+ st
x = x' & y = [0,y'] & one = [0,x' *' y'] by A69,A70,Def3;
hence y = z by Th7;
suppose x in [:{0},REAL+:] & y in REAL+;
then ex x',y' being Element of REAL+ st
x = [0,x'] & y = y' & one = [0,y' *' x'] by A70,A77,Def3;
hence y = z by Th7;
suppose x in [:{0},REAL+:] & z in REAL+;
then ex x',z' being Element of REAL+ st
x = [0,x'] & z = z' & one = [0,z' *' x'] by A71,A78,Def3;
hence y = z by Th7;
suppose x in REAL+ & z in [:{0},REAL+:];
then ex x',z' being Element of REAL+ st
x = x' & z = [0,z'] & one = [0,x' *' z'] by A69,A71,Def3;
hence y = z by Th7;
suppose not (x in REAL+ & y in REAL+) &
not (x in REAL+ & y in [:{0},REAL+:] & x <> 0) &
not (y in REAL+ & x in [:{0},REAL+:] & y <> 0) &
not (x in [:{0},REAL+:] & y in [:{0},REAL+:]);
hence thesis by A70,Def3;
suppose not (x in REAL+ & z in REAL+) &
not (x in REAL+ & z in [:{0},REAL+:] & x <> 0) &
not (z in REAL+ & x in [:{0},REAL+:] & z <> 0) &
not (x in [:{0},REAL+:] & z in [:{0},REAL+:]);
hence thesis by A71,Def3;
end;
thus thesis;
end;
consistency;
involutiveness
proof let x,y be Element of REAL;
assume that
A89:  y <> 0 implies *(y,x) = one and
A90:  y = 0 implies x = 0;
thus x <> 0 implies *(x,y) = one by A89,A90;
assume
A91:  x = 0;
assume
A92:   y <> 0;
A93:    y in REAL+ \/ [:{0},REAL+:] by NUMBERS:def 1,XBOOLE_0:def 4;
per cases by A93,XBOOLE_0:def 2;
suppose y in REAL+;
then ex x',y' being Element of REAL+
st x = x' & y = y' & one = x' *' y' by A89,A91,A92,Def3,ARYTM_2:21;
suppose y in [:{0},REAL+:];
then not y in REAL+ & not x in [:{0},REAL+:]
by A91,Th5,ARYTM_2:21,XBOOLE_0:3;
end;
end;

begin :: from COMPLEX1

Lm3:
for x,y,z being set st [x,y] = {z}
holds z = {x} & x = y
proof let x,y,z be set;
assume [x,y] = {z};
then {{x,y},{x}} = {z} by TARSKI:def 5;
then A1:  {x,y} in {z} & {x} in {z} by TARSKI:def 2;
hence
A2: z = {x} by TARSKI:def 1;
{x,y} = z by A1,TARSKI:def 1;
hence x = y by A2,ZFMISC_1:9;
end;

reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;

theorem Th10:
not (0,one)-->(a,b) in REAL
proof set f = (0,one)-->(a,b);
A1: f = {[0,a],[one,b]} by FUNCT_4:71;
now assume f in {[i,j]: i,j are_relative_prime & j <> {}};
then consider i,j such that
A2:   f = [i,j] and
i,j are_relative_prime and
j <> {};
A3:   f = {{i,j},{i}} by A2,TARSKI:def 5;
then A4:    {i} in f by TARSKI:def 2;
A5:  {i,j} in f by A3,TARSKI:def 2;
A6:  [0,a] = {{0,a},{0}} by TARSKI:def 5;
A7:  [one,b] = {{one,b},{one}} by TARSKI:def 5;
A8: now assume i = j;
then {i} = {i,j} by ENUMSET1:69;
then [i,j] = {{i}} by A2,A3,ENUMSET1:69;
then [0,a] in {{i}} & [one,b] in {{i}} by A1,A2,TARSKI:def 2;
then [0,a] = {i} & [one,b] = {i} by TARSKI:def 1;
end;
per cases by A1,A4,A5,TARSKI:def 2;
suppose {i,j} = [0,a] & {i} = [0,a];
suppose that
A9:  {i,j} = [0,a] and
A10:  {i} = [one,b];
A11:    i = {one} by A10,Lm3;
i in [0,a] by A9,TARSKI:def 2;
then i = {0,a} or i = {0} by A6,TARSKI:def 2;
then 0 in i by TARSKI:def 1,def 2;
suppose that
A12:  {i,j} = [one,b] and
A13:  {i} = [0,a];
A14:    i = {0} by A13,Lm3;
i in [one,b] by A12,TARSKI:def 2;
then i = {one,b} or i = {one} by A7,TARSKI:def 2;
then one in i by TARSKI:def 1,def 2;
suppose {i,j} = [one,b] & {i} = [one,b];
end;
then A15: not f in
{[i,j]: i,j are_relative_prime & j <> {}} \ {[k,one]: not contradiction}
by XBOOLE_0:def 4;
now assume f in omega;
then f is ordinal by ORDINAL1:23;
then {} in f by A1,ORDINAL3:10;
end;
then A16: not f in RAT+ by A15,ARYTM_3:def 6,XBOOLE_0:def 2;
set IR = { A where A is Subset of RAT+:
for r being Element of RAT+ st r in A holds
(for s being Element of RAT+ st s <=' r holds s in A) &
ex s being Element of RAT+ st s in A & r < s};
not ex x,y being set st {[0,x],[one,y]} in IR
proof given x,y being set such that
A17: {[0,x],[one,y]} in IR;
consider A being Subset of RAT+ such that
A18: {[0,x],[one,y]} = A and
A19: for r being Element of RAT+ st r in A holds
(for s being Element of RAT+ st s <=' r holds s in A) &
ex s being Element of RAT+ st s in A & r < s by A17;
A20: [0,x] in A by A18,TARSKI:def 2;
for r,s being Element of RAT+ st r in A & s <=' r holds s in A by A19;
then consider r1,r2,r3 being Element of RAT+ such that
A21: r1 in A & r2 in A & r3 in A and
A22: r1 <> r2 & r2 <> r3 & r3 <> r1 by A20,ARYTM_3:82;
(r1 = [0,x] or r1 = [one,y]) &
(r2 = [0,x] or r2 = [one,y]) &
(r3 = [0,x] or r3 = [one,y]) by A18,A21,TARSKI:def 2;
end;
then not f in IR by A1;
then not f in DEDEKIND_CUTS by ARYTM_2:def 1,XBOOLE_0:def 4;
then not f in RAT+ \/ DEDEKIND_CUTS by A16,XBOOLE_0:def 2;
then A23: not f in REAL+ by ARYTM_2:def 2,XBOOLE_0:def 4;
now assume f in [:{{}},REAL+:];
then consider x,y being set such that
A24:  x in {{}} and
y in REAL+ and
A25:  f = [x,y] by ZFMISC_1:103;
x = 0 by A24,TARSKI:def 1;
then A26:    f = {{0,y},{0}} by A25,TARSKI:def 5;
f = {[0,a],[one,b]} by FUNCT_4:71;
then [one,b] in f by TARSKI:def 2;
then A27:   [one,b] = {0} or [one,b] = {0,y} by A26,TARSKI:def 2;
per cases by A27,TARSKI:def 5;
suppose {{one,b},{one}} = {0};
then 0 in {{one,b},{one}} by TARSKI:def 1;
suppose {{one,b},{one}} = {0,y};
then 0 in {{one,b},{one}} by TARSKI:def 2;
end;
then not f in REAL+ \/ [:{{}},REAL+:] by A23,XBOOLE_0:def 2;
hence thesis by NUMBERS:def 1,XBOOLE_0:def 4;
end;

definition let x,y be Element of REAL;
canceled;
func [*x,y*] -> Element of COMPLEX equals
:Def7:  x if y = 0
otherwise (0,one) --> (x,y);
consistency;
coherence
proof
thus y = 0 implies x is Element of COMPLEX by NUMBERS:def 2,XBOOLE_0:def 2;
set z = (0,one)-->(x,y);
assume
A1:  y <> 0;
A2:  z in Funcs({0,one},REAL) by FUNCT_2:11;
now assume z in { r where r is Element of Funcs({0,one},REAL): r.one = 0
}
;
then ex r being Element of Funcs({0,one},REAL) st z = r & r.one = 0;
end;
then z in
Funcs({0,one},REAL)
\ { r where r is Element of Funcs({0,one},REAL): r.one = 0}
by A2,XBOOLE_0:def 4;
hence thesis by NUMBERS:def 2,XBOOLE_0:def 2;
end;
end;

theorem
for c being Element of COMPLEX
ex r,s being Element of REAL st c = [*r,s*]
proof let c be Element of COMPLEX;
per cases;
suppose c in REAL;
then reconsider r=c, z=0 as Element of REAL by Th1,ARYTM_2:21;
take r,z;
thus c = [*r,z*] by Def7;
suppose not c in REAL;
then A1:  c in Funcs({0,one},REAL)
\ { x where x is Element of Funcs({0,one},REAL): x.one = 0}
by NUMBERS:def 2,XBOOLE_0:def 2;
then A2: c in Funcs({0,one},REAL) by XBOOLE_0:def 4;
then consider f being Function such that
A3: c = f and
A4: dom f = {0,one} and
A5: rng f c= REAL by FUNCT_2:def 2;
0 in {0,one} & one in {0,one} by TARSKI:def 2;
then f.0 in rng f & f.one in rng f by A4,FUNCT_1:12;
then reconsider r = f.0, s = f.one as Element of REAL by A5;
A6: c = (0,one)-->(r,s) by A3,A4,FUNCT_4:69;
take r,s;
now assume s = 0;
then (0,one)-->(r,s).one = 0 by FUNCT_4:66;
then c in { x where x is Element of Funcs({0,one},REAL): x.one = 0}
by A2,A6;
end;
hence c = [*r,s*] by A6,Def7;
end;

theorem
for x1,x2,y1,y2 being Element of REAL st [*x1,x2*] = [*y1,y2*]
holds x1 = y1 & x2 = y2
proof let x1,x2,y1,y2 be Element of REAL such that
A1: [*x1,x2*] = [*y1,y2*];
per cases;
suppose
A2: x2 = 0;
then A3: [*x1,x2*] = x1 by Def7;
A4: now assume y2 <> 0;
then x1 = (0,one) --> (y1,y2) by A1,A3,Def7;
end;
hence x1 = y1 by A1,A3,Def7;
thus x2 = y2 by A2,A4;
suppose x2 <> 0;
then A5: [*y1,y2*] = (0,one) --> (x1,x2) by A1,Def7;
now assume y2 = 0;
then [*y1,y2*] = y1 by Def7;
end;
then [*y1,y2*] = (0,one) --> (y1,y2) by Def7;
hence x1 = y1 & x2 = y2 by A5,FUNCT_4:72;
end;

set RR = [:{0},REAL+:] \ {[0,0]};

reconsider o = 0 as Element of REAL by Th1,ARYTM_2:21;

theorem Th13:
for x,o being Element of REAL st o = 0 holds +(x,o) = x
proof let x,o being Element of REAL such that
A1: o = 0;
reconsider y' = 0 as Element of REAL+ by ARYTM_2:21;
per cases;
suppose x in REAL+;
then reconsider x' = x as Element of REAL+;
x' = x' + y' by ARYTM_2:def 8;
hence +(x,o) = x by A1,Def2;
suppose
A2:  not x in REAL+;
x in REAL+ \/ [:{{}},REAL+:] by NUMBERS:def 1,XBOOLE_0:def 4;
then A3: x in [:{{}},REAL+:] by A2,XBOOLE_0:def 2;
then consider x1,x2 being set such that
A4: x1 in {{}} and
A5: x2 in REAL+ and
A6: x = [x1,x2] by ZFMISC_1:103;
reconsider x' = x2 as Element of REAL+ by A5;
A7:  x1 = 0 by A4,TARSKI:def 1;
then x' <> 0 by A6,Th3;
then x = y' - x' by A6,A7,ARYTM_1:19;
hence thesis by A1,A3,A6,A7,Def2;
end;

theorem Th14:
for x,o being Element of REAL st o = 0 holds *(x,o) = 0
proof let x,o being Element of REAL such that
A1: o = 0;
per cases;
suppose x in REAL+;
then reconsider x' = x, y' = 0 as Element of REAL+ by ARYTM_2:21;
0 = x' *' y' by ARYTM_2:4;
hence *(x,o) = 0 by A1,Def3;
suppose
A2: not x in REAL+;
not o in [:{{}},REAL+:] by A1,Th5,ARYTM_2:21,XBOOLE_0:3;
hence thesis by A1,A2,Def3;
end;

theorem Th15:
for x,y,z being Element of REAL holds *(x,*(y,z)) = *(*(x,y),z)
proof let x,y,z be Element of REAL;
per cases;
suppose that
A1: x in REAL+ and
A2: y in REAL+ and
A3: z in REAL+;
consider y',z' being Element of REAL+ such that
A4: y = y' and
A5: z = z' and
A6: *(y,z) = y' *' z' by A2,A3,Def3;
consider x',yz' being Element of REAL+ such that
A7: x = x' and
A8: *(y,z) = yz' and
A9: *(x,*(y,z)) = x' *' yz' by A1,A6,Def3;
consider x'',y'' being Element of REAL+ such that
A10: x = x'' and
A11: y = y'' and
A12: *(x,y) = x'' *' y'' by A1,A2,Def3;
consider xy'',z'' being Element of REAL+ such that
A13: *(x,y) = xy'' and
A14: z = z'' and
A15: *(*(x,y),z) = xy'' *' z'' by A3,A12,Def3;
thus *(x,*(y,z)) = *(*(x,y),z) by A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,
ARYTM_2:13;
suppose that
A16: x in REAL+ & x <> 0 and
A17: y in RR and
A18: z in REAL+ & z <> 0;
A19: y in [:{0},REAL+:] by A17,XBOOLE_0:def 4;
then consider y',z' being Element of REAL+ such that
A20: y = [0,y'] and
A21: z = z' and
A22: *(y,z) = [0,z' *' y'] by A18,Def3;
*(y,z) in [:{0},REAL+:] by A22,Lm1,ZFMISC_1:106;
then consider x',yz' being Element of REAL+ such that
A23: x = x' and
A24: *(y,z) = [0,yz'] and
A25: *(x,*(y,z) ) = [0,x' *' yz'] by A16,Def3;
consider x'',y'' being Element of REAL+ such that
A26: x = x'' and
A27: y = [0,y''] and
A28: *(x,y) = [0,x'' *' y''] by A16,A19,Def3;
*(x,y) in [:{0},REAL+:] by A28,Lm1,ZFMISC_1:106;
then consider xy'',z'' being Element of REAL+ such that
A29: *(x,y) = [0,xy''] and
A30: z = z'' and
A31: *(*(x,y),z) = [0,z'' *' xy''] by A18,Def3;
A32: y' = y'' by A20,A27,ZFMISC_1:33;
thus *(x,*(y,z)) = [0,x' *' (y' *' z')] by A22,A24,A25,ZFMISC_1:33
.= [0,(x'' *' y'') *' z''] by A21,A23,A26,A30,A32,ARYTM_2:13
.= *(*(x,y),z) by A28,A29,A31,ZFMISC_1:33;
suppose that
A33: x in RR and
A34: y in REAL+ & y <> 0 and
A35: z in REAL+ & z <> 0;
A36: x in [:{0},REAL+:] by A33,XBOOLE_0:def 4;
consider y',z' being Element of REAL+ such that
A37: y = y' and
A38: z = z' and
A39: *(y,z) = y' *' z' by A34,A35,Def3;
y' *' z' <> 0 by A34,A35,A37,A38,ARYTM_1:2;
then consider x',yz' being Element of REAL+ such that
A40: x = [0,x'] and
A41: *(y,z) = yz' and
A42: *(x,*(y,z)) = [0,yz' *' x'] by A36,A39,Def3;
consider x'',y'' being Element of REAL+ such that
A43: x = [0,x''] and
A44: y = y'' and
A45: *(x,y) = [0,y'' *' x''] by A34,A36,Def3;
*(x,y) in [:{0},REAL+:] by A45,Lm1,ZFMISC_1:106;
then consider xy'',z'' being Element of REAL+ such that
A46: *(x,y) = [0,xy''] and
A47: z = z'' and
A48: *(*(x,y),z) = [0,z'' *' xy''] by A35,Def3;
x' = x'' by A40,A43,ZFMISC_1:33;
hence *(x,*(y,z)) = [0,(x'' *' y'') *' z''] by A37,A38,A39,A41,A42,A44,A47,
ARYTM_2:13
.= *(*(x,y),z) by A45,A46,A48,ZFMISC_1:33;
suppose that
A49: x in RR and
A50: y in RR and
A51: z in REAL+ & z <> 0;
A52: x in [:{0},REAL+:] by A49,XBOOLE_0:def 4;
A53: y in [:{0},REAL+:] by A50,XBOOLE_0:def 4;
then consider y',z' being Element of REAL+ such that
A54: y = [0,y'] and
A55: z = z' and
A56: *(y,z) = [0,z' *' y'] by A51,Def3;
*(y,z) in [:{0},REAL+:] by A56,Lm1,ZFMISC_1:106;
then consider x',yz' being Element of REAL+ such that
A57: x = [0,x'] and
A58: *(y,z) = [0,yz'] and
A59: *(x,*(y,z)) = yz' *' x' by A52,Def3;
consider x'',y'' being Element of REAL+ such that
A60: x = [0,x''] and
A61: y = [0,y''] and
A62: *(x,y) = y'' *' x'' by A52,A53,Def3;
consider xy'',z'' being Element of REAL+ such that
A63: *(x,y) = xy'' and
A64: z = z'' and
A65: *(*(x,y),z) = xy'' *' z'' by A51,A62,Def3;
A66: x' = x'' by A57,A60,ZFMISC_1:33;
A67: y' = y'' by A54,A61,ZFMISC_1:33;
thus *(x,*(y,z)) = x' *' (y' *' z') by A56,A58,A59,ZFMISC_1:33
.= *(*(x,y),z) by A55,A62,A63,A64,A65,A66,A67,ARYTM_2:13;
suppose that
A68: x in REAL+ & x <> 0 and
A69: y in REAL+ & y <> 0 and
A70: z in RR;
A71: z in [:{0},REAL+:] by A70,XBOOLE_0:def 4;
then consider y',z' being Element of REAL+ such that
A72: y = y' and
A73: z = [0,z'] and
A74: *(y,z) = [0,y' *' z'] by A69,Def3;
*(y,z) in [:{0},REAL+:] by A74,Lm1,ZFMISC_1:106;
then consider x',yz' being Element of REAL+ such that
A75: x = x' and
A76: *(y,z) = [0,yz'] and
A77: *(x,*(y,z)) = [0,x' *' yz'] by A68,Def3;
consider x'',y'' being Element of REAL+ such that
A78: x = x'' and
A79: y = y'' and
A80: *(x,y) = x'' *' y'' by A68,A69,Def3;
*(x,y) <> 0 by A68,A69,A78,A79,A80,ARYTM_1:2;
then consider xy'',z'' being Element of REAL+ such that
A81: *(x,y) = xy'' and
A82: z = [0,z''] and
A83: *(*(x,y),z) = [0,xy'' *' z''] by A71,A80,Def3;
A84: z' = z'' by A73,A82,ZFMISC_1:33;
thus *(x,*(y,z)) = [0,x' *' (y' *' z')] by A74,A76,A77,ZFMISC_1:33
.= *(*(x,y),z) by A72,A75,A78,A79,A80,A81,A83,A84,ARYTM_2:13;
suppose that
A85: x in REAL+ & x <> 0 and
A86: y in RR and
A87: z in RR;
A88: y in [:{0},REAL+:] by A86,XBOOLE_0:def 4;
A89: z in [:{0},REAL+:] by A87,XBOOLE_0:def 4;
then consider y',z' being Element of REAL+ such that
A90: y = [0,y'] and
A91: z = [0,z'] and
A92: *(y,z) = z' *' y' by A88,Def3;
consider x',yz' being Element of REAL+ such that
A93: x = x' and
A94: *(y,z) = yz' and
A95: *(x,*(y,z)) = x' *' yz' by A85,A92,Def3;
consider x'',y'' being Element of REAL+ such that
A96: x = x'' and
A97: y = [0,y''] and
A98: *(x,y) = [0,x'' *' y''] by A85,A88,Def3;
*(x,y) in [:{0},REAL+:] by A98,Lm1,ZFMISC_1:106;
then consider xy'',z'' being Element of REAL+ such that
A99: *(x,y) = [0,xy''] and
A100: z = [0,z''] and
A101: *(*(x,y),z) = z'' *' xy'' by A89,Def3;
A102: y' = y'' by A90,A97,ZFMISC_1:33;
z' = z'' by A91,A100,ZFMISC_1:33;
hence *(x,*(y,z)) = (x'' *' y'') *' z'' by A92,A93,A94,A95,A96,A102,ARYTM_2:13
.= *(*(x,y),z) by A98,A99,A101,ZFMISC_1:33;
suppose that
A103: y in REAL+ & y <> 0 and
A104: x in RR and
A105: z in RR;
A106: x in [:{0},REAL+:] by A104,XBOOLE_0:def 4;
A107: z in [:{0},REAL+:] by A105,XBOOLE_0:def 4;
then consider y',z' being Element of REAL+ such that
A108: y = y' and
A109: z = [0,z'] and
A110: *(y,z) = [0,y' *' z'] by A103,Def3;
[0,y' *' z'] in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider x',yz' being Element of REAL+ such that
A111: x = [0,x'] and
A112: *(y,z) = [0,yz'] and
A113: *(x,*(y,z)) = yz' *' x' by A106,A110,Def3;
consider x'',y'' being Element of REAL+ such that
A114: x = [0,x''] and
A115: y = y'' and
A116: *(x,y) = [0,y'' *' x''] by A103,A106,Def3;
*(x,y) in [:{0},REAL+:] by A116,Lm1,ZFMISC_1:106;
then consider xy'',z'' being Element of REAL+ such that
A117: *(x,y) = [0,xy''] and
A118: z = [0,z''] and
A119: *(*(x,y),z) = z'' *' xy'' by A107,Def3;
A120: x' = x'' by A111,A114,ZFMISC_1:33;
A121: z' = z'' by A109,A118,ZFMISC_1:33;
thus *(x,*(y,z)) = x' *' (y' *' z') by A110,A112,A113,ZFMISC_1:33
.= (x'' *' y'') *' z'' by A108,A115,A120,A121,ARYTM_2:13
.= *(*(x,y),z) by A116,A117,A119,ZFMISC_1:33;
suppose that
A122: x in RR and
A123: y in RR and
A124: z in RR;
A125: x in [:{0},REAL+:] by A122,XBOOLE_0:def 4;
A126: y in [:{0},REAL+:] by A123,XBOOLE_0:def 4;
A127: z in [:{0},REAL+:] by A124,XBOOLE_0:def 4;
then consider y',z' being Element of REAL+ such that
A128: y = [0,y'] and
A129: z = [0,z'] and
A130: *(y,z) = z' *' y' by A126,Def3;
not z in {[0,0]} & not y in {[0,0]} by A123,A124,XBOOLE_0:def 4;
then z' <> 0 & y' <> 0 by A128,A129,TARSKI:def 1;
then *(z,y) <> 0 by A130,ARYTM_1:2;
then consider x',yz' being Element of REAL+ such that
A131: x = [0,x'] and
A132: *(y,z) = yz' and
A133: *(x,*(y,z)) = [0,yz' *' x'] by A125,A130,Def3;
consider x'',y'' being Element of REAL+ such that
A134: x = [0,x''] and
A135: y = [0,y''] and
A136: *(x,y) = y'' *' x'' by A125,A126,Def3;
A137: x' = x'' by A131,A134,ZFMISC_1:33;
A138: y' = y'' by A128,A135,ZFMISC_1:33;
not x in {[0,0]} & not y in {[0,0]} by A122,A123,XBOOLE_0:def 4;
then x' <> 0 & y' <> 0 by A128,A131,TARSKI:def 1;
then *(x,y) <> 0 by A136,A137,A138,ARYTM_1:2;
then consider xy'',z'' being Element of REAL+ such that
A139: *(x,y) = xy'' and
A140: z = [0,z''] and
A141: *(*(x,y),z) = [0,xy'' *' z''] by A127,A136,Def3;
z' = z'' by A129,A140,ZFMISC_1:33;
hence *(x,*(y,z)) = *(*(x,y),z) by A130,A132,A133,A136,A137,A138,A139,A141,
ARYTM_2:13;
suppose
A142: x = 0;
hence *(x,*(y,z)) = 0 by Th14
.= *(o,z) by Th14
.= *(*(x,y),z) by A142,Th14;
suppose
A143:  y = 0;
hence *(x,*(y,z)) = *(x,o) by Th14
.= 0 by Th14
.= *(o,z) by Th14
.= *(*(x,y),z) by A143,Th14;
suppose
A144: z = 0;
hence *(x,*(y,z)) = *(x,o) by Th14
.= 0 by Th14
.= *(*(x,y),z) by A144,Th14;
suppose
A145: not( x in REAL+ & y in REAL+ & z in REAL+) &
not(x in REAL+ & y in RR & z in REAL+) &
not(y in REAL+ & x in RR & z in REAL+) &
not(x in RR & y in RR & z in REAL+) &
not( x in REAL+ & y in REAL+ & z in RR) &
not(x in REAL+ & y in RR & z in RR) &
not(y in REAL+ & x in RR & z in RR) &
not(x in RR & y in RR & z in RR);
A146: not [0,0] in REAL+ by ARYTM_2:3;
REAL = (REAL+ \ {[{},{}]}) \/ ([:{{}},REAL+:] \ {[{},{}]})
by NUMBERS:def 1,XBOOLE_1:42
.= REAL+ \/ RR by A146,ZFMISC_1:65;
hence thesis by A145,XBOOLE_0:def 2;
end;

theorem Th16:
for x,y,z being Element of REAL holds *(x,+(y,z)) = +(*(x,y),*(x,z))
proof let x,y,z be Element of REAL;
per cases;
suppose
A1: x = 0;
hence *(x,+(y,z)) = 0 by Th14
.= +(o,o) by Th13
.= +(*(x,y),o) by A1,Th14
.= +(*(x,y),*(x,z)) by A1,Th14;
suppose that
A2: x in REAL+ and
A3: y in REAL+ and
A4: z in REAL+;
consider x',y' being Element of REAL+ such that
A5: x = x' and
A6: y = y' and
A7: *(x,y) = x' *' y' by A2,A3,Def3;
consider x'',z' being Element of REAL+ such that
A8: x = x'' and
A9: z = z' and
A10: *(x,z) = x'' *' z' by A2,A4,Def3;
consider xy',xz' being Element of REAL+ such that
A11: *(x,y) = xy' and
A12: *(x,z) = xz' and
A13: +(*(x,y),*(x,z)) = xy' + xz' by A7,A10,Def2;
consider y'',z'' being Element of REAL+ such that
A14: y = y'' and
A15: z = z'' and
A16: +(y,z) = y'' + z'' by A3,A4,Def2;
consider x''',yz' being Element of REAL+ such that
A17: x = x''' and
A18: +(y,z) = yz' and
A19: *(x,+(y,z)) = x''' *' yz' by A2,A16,Def3;
thus *(x,+(y,z)) = +(*(x,y),*(x,z)) by A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,
A16,A17,A18,A19,ARYTM_2:14;
suppose that
A20: x in REAL+ & x <> 0 and
A21: y in REAL+ and
A22: z in RR;
A23: z in [:{0},REAL+:] by A22,XBOOLE_0:def 4;
consider x',y' being Element of REAL+ such that
A24: x = x' and
A25: y = y' and
A26: *(x,y) = x' *' y' by A20,A21,Def3;
consider x'',z' being Element of REAL+ such that
A27: x = x'' and
A28: z = [0,z'] and
A29: *(x,z) = [0,x'' *' z'] by A20,A23,Def3;
consider y'',z'' being Element of REAL+ such that
A30: y = y'' and
A31: z = [0,z''] and
A32: +(y,z) = y'' - z'' by A21,A23,Def2;
A33: z' = z'' by A28,A31,ZFMISC_1:33;
*(x,z) in [:{0},REAL+:] by A29,Lm1,ZFMISC_1:106;
then consider xy',xz' being Element of REAL+ such that
A34: *(x,y) = xy' and
A35: *(x,z) = [0,xz'] and
A36: +(*(x,y),*(x,z)) = xy' - xz' by A26,Def2;
now per cases;
suppose
A37:   z'' <=' y'';
then A38:   +(y,z) = y'' -' z'' by A32,ARYTM_1:def 2;
then consider x''',yz' being Element of REAL+ such that
A39: x = x''' and
A40: +(y,z) = yz' and
A41: *(x,+(y,z)) = x''' *' yz' by A20,Def3;
thus *(x,+(y,z)) = (x' *' y') - (x'' *' z') by A24,A25,A27,A30,A33,A37,A38,
A39,A40,A41,ARYTM_1:26
.= +(*(x,y),*(x,z)) by A26,A29,A34,A35,A36,ZFMISC_1:33;
suppose
A42:  not z'' <=' y'';
then A43:   +(y,z) = [0,z'' -' y''] by A32,ARYTM_1:def 2;
then +(y,z) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider x''',yz' being Element of REAL+ such that
A44:  x = x''' and
A45:  +(y,z) = [0,yz'] and
A46:  *(x,+(y,z)) = [0,x''' *' yz'] by A20,Def3;
thus *(x,+(y,z)) = [0,x''' *' (z'' -' y'')] by A43,A45,A46,ZFMISC_1:33
.= (x' *' y') - (x'' *'
z') by A20,A24,A25,A27,A30,A33,A42,A44,ARYTM_1:27
.= +(*(x,y),*(x,z)) by A26,A29,A34,A35,A36,ZFMISC_1:33;
end;
hence thesis;
suppose that
A47: x in REAL+ & x <> 0 and
A48: z in REAL+ and
A49: y in RR;
A50: y in [:{0},REAL+:] by A49,XBOOLE_0:def 4;
consider x',z' being Element of REAL+ such that
A51: x = x' and
A52: z = z' and
A53: *(x,z) = x' *' z' by A47,A48,Def3;
consider x'',y' being Element of REAL+ such that
A54: x = x'' and
A55: y = [0,y'] and
A56: *(x,y) = [0,x'' *' y'] by A47,A50,Def3;
consider z'',y'' being Element of REAL+ such that
A57: z = z'' and
A58: y = [0,y''] and
A59: +(z,y) = z'' - y'' by A48,A50,Def2;
A60: y' = y'' by A55,A58,ZFMISC_1:33;
*(x,y) in [:{0},REAL+:] by A56,Lm1,ZFMISC_1:106;
then consider xz',xy' being Element of REAL+ such that
A61: *(x,z) = xz' and
A62: *(x,y) = [0,xy'] and
A63: +(*(x,z),*(x,y)) = xz' - xy' by A53,Def2;
now per cases;
suppose
A64:   y'' <=' z'';
then A65:   +(z,y) = z'' -' y'' by A59,ARYTM_1:def 2;
then consider x''',zy' being Element of REAL+ such that
A66: x = x''' and
A67: +(z,y) = zy' and
A68: *(x,+(z,y)) = x''' *' zy' by A47,Def3;
thus *(x,+(z,y)) = (x' *' z') - (x'' *' y') by A51,A52,A54,A57,A60,A64,A65,
A66,A67,A68,ARYTM_1:26
.= +(*(x,z),*(x,y)) by A53,A56,A61,A62,A63,ZFMISC_1:33;
suppose
A69:  not y'' <=' z'';
then A70:   +(z,y) = [0,y'' -' z''] by A59,ARYTM_1:def 2;
then +(z,y) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider x''',zy' being Element of REAL+ such that
A71:  x = x''' and
A72:  +(z,y) = [0,zy'] and
A73:  *(x,+(z,y)) = [0,x''' *' zy'] by A47,Def3;
thus *(x,+(z,y)) = [0,x''' *' (y'' -' z'')] by A70,A72,A73,ZFMISC_1:33
.= (x' *' z') - (x'' *' y')
by A47,A51,A52,A54,A57,A60,A69,A71,ARYTM_1:27
.= +(*(x,z),*(x,y)) by A53,A56,A61,A62,A63,ZFMISC_1:33;
end;
hence thesis;
suppose that
A74: x in REAL+ & x <> 0 and
A75: y in RR and
A76: z in RR;
A77: y in [:{0},REAL+:] by A75,XBOOLE_0:def 4;
A78: z in [:{0},REAL+:] by A76,XBOOLE_0:def 4;
consider x',y' being Element of REAL+ such that
A79: x = x' and
A80: y = [0,y'] and
A81: *(x,y) = [0,x' *' y'] by A74,A77,Def3;
consider x'',z' being Element of REAL+ such that
A82: x = x'' and
A83: z = [0,z'] and
A84: *(x,z) = [0,x'' *' z'] by A74,A78,Def3;
*(x,y) in [:{0},REAL+:] & *(x,z) in [:{0},REAL+:]
by A81,A84,Lm1,ZFMISC_1:106;
then not(*(x,y) in REAL+ & *(x,z) in REAL+) &
not(*(x,y) in REAL+ & *(x,z) in [:{0},REAL+:]) &
not(*(x,y) in [:{0},REAL+:] & *(x,z) in REAL+) by Th5,XBOOLE_0:3;
then consider xy',xz' being Element of REAL+ such that
A85: *(x,y) = [0,xy'] and
A86: *(x,z) = [0,xz'] and
A87: +(*(x,y),*(x,z)) = [0,xy' + xz'] by Def2;
not(y in REAL+ & z in REAL+) &
not(y in REAL+ & z in [:{0},REAL+:]) &
not(y in [:{0},REAL+:] & z in REAL+) by A77,A78,Th5,XBOOLE_0:3;
then consider y'',z'' being Element of REAL+ such that
A88: y = [0,y''] and
A89: z = [0,z''] and
A90: +(y,z) = [0,y'' + z''] by Def2;
+(y,z) in [:{0},REAL+:] by A90,Lm1,ZFMISC_1:106;
then consider x''',yz' being Element of REAL+ such that
A91: x = x''' and
A92: +(y,z) = [0,yz'] and
A93: *(x,+(y,z)) = [0,x''' *' yz'] by A74,Def3;
A94: xy' = x' *' y' by A81,A85,ZFMISC_1:33;
A95: y' = y'' by A80,A88,ZFMISC_1:33;
A96: z' = z'' by A83,A89,ZFMISC_1:33;
thus *(x,+(y,z)) = [0,x''' *' (y'' + z'')] by A90,A92,A93,ZFMISC_1:33
.= [0,(x' *' y') + (x' *' z')] by A79,A91,A95,A96,ARYTM_2:14
.= +(*(x,y),*(x,z)) by A79,A82,A84,A86,A87,A94,ZFMISC_1:33;
suppose that
A97: x in RR and
A98: y in REAL+ and
A99: z in REAL+;
A100: x in [:{0},REAL+:] by A97,XBOOLE_0:def 4;
consider y'',z'' being Element of REAL+ such that
A101: y = y'' and
A102: z = z'' and
A103: +(y,z) = y'' + z'' by A98,A99,Def2;
now per cases;
suppose that
A104: y <> 0 and
A105: z <> 0;
consider x',y' being Element of REAL+ such that
A106: x = [0,x'] and
A107: y = y' and
A108: *(x,y) = [0,y' *' x'] by A98,A100,A104,Def3;
consider x'',z' being Element of REAL+ such that
A109: x = [0,x''] and
A110: z = z' and
A111: *(x,z) = [0,z' *' x''] by A99,A100,A105,Def3;
*(x,y) in [:{0},REAL+:] & *(x,z) in [:{0},REAL+:]
by A108,A111,Lm1,ZFMISC_1:106;
then not(*(x,y) in REAL+ & *(x,z) in REAL+) &
not(*(x,y) in REAL+ & *(x,z) in [:{0},REAL+:]) &
not(*(x,y) in [:{0},REAL+:] & *(x,z) in REAL+) by Th5,XBOOLE_0:3;
then consider xy',xz' being Element of REAL+ such that
A112: *(x,y) = [0,xy'] and
A113: *(x,z) = [0,xz'] and
A114: +(*(x,y),*(x,z)) = [0,xy' + xz'] by Def2;
y'' + z'' <> 0 by A102,A105,ARYTM_2:6;
then consider x''',yz' being Element of REAL+ such that
A115: x = [0,x'''] and
A116: +(y,z) = yz' and
A117: *(x,+(y,z)) = [0,yz' *' x'''] by A100,A103,Def3;
A118: xy' = x' *' y' by A108,A112,ZFMISC_1:33;
A119: x' = x'' by A106,A109,ZFMISC_1:33;
x'' = x''' by A109,A115,ZFMISC_1:33;
hence *(x,+(y,z)) = [0,(x' *' y') + (x'' *' z')] by A101,A102,A103,A107,A110,
A116,A117,A119,ARYTM_2:14
.= +(*(x,y),*(x,z)) by A111,A113,A114,A118,ZFMISC_1:33;
suppose
A120: x = 0;
hence *(x,+(y,z)) = 0 by Th14
.= +(o,o) by Th13
.= +(*(x,y),o) by A120,Th14
.= +(*(x,y),*(x,z)) by A120,Th14;
suppose
A121: z = 0;
hence *(x,+(y,z)) = *(x,y) by Th13
.= +(*(x,y),o) by Th13
.= +(*(x,y),*(x,z)) by A121,Th14;
suppose
A122: y = 0;
hence *(x,+(y,z)) = *(x,z) by Th13
.= +(o,*(x,z)) by Th13
.= +(*(x,y),*(x,z)) by A122,Th14;
end;
hence thesis;
suppose that
A123: x in RR and
A124: y in REAL+ and
A125: z in RR;
A126: x in [:{0},REAL+:] by A123,XBOOLE_0:def 4;
A127: z in [:{0},REAL+:] by A125,XBOOLE_0:def 4;
then consider x'',z' being Element of REAL+ such that
A128: x = [0,x''] and
A129: z = [0,z'] and
A130: *(x,z) = z' *' x'' by A126,Def3;
now per cases;
suppose y <> 0;
then consider x',y' being Element of REAL+ such that
A131: x = [0,x'] and
A132: y = y' and
A133: *(x,y) = [0,y' *' x'] by A124,A126,Def3;
consider y'',z'' being Element of REAL+ such that
A134: y = y'' and
A135: z = [0,z''] and
A136: +(y,z) = y'' - z'' by A127,A132,Def2;
*(x,y) in [:{0},REAL+:] by A133,Lm1,ZFMISC_1:106;
then consider xy',xz' being Element of REAL+ such that
A137: *(x,y) = [0,xy'] and
A138: *(x,z) = xz' and
A139: +(*(x,y),*(x,z)) = xz' - xy' by A130,Def2;
A140: z' = z'' by A129,A135,ZFMISC_1:33;
now per cases;
suppose
A141: z'' <=' y'';
then A142: +(y,z) = y'' -' z'' by A136,ARYTM_1:def 2;
now per cases;
suppose
A143: +(y,z) <> 0;
then consider x''',yz' being Element of REAL+ such that
A144: x = [0,x'''] and
A145: +(y,z) = yz' and
A146: *(x,+(y,z)) = [0,yz' *' x'''] by A126,A142,Def3;
A147: x' = x'' by A128,A131,ZFMISC_1:33;
A148: x'' = x''' by A128,A144,ZFMISC_1:33;
A149: z' = z'' by A129,A135,ZFMISC_1:33;
not x in {[0,0]} by NUMBERS:def 1,XBOOLE_0:def 4;
then x''' <> 0 by A144,TARSKI:def 1;
hence *(x,+(y,z)) = (x' *' z') - (x' *' y') by A132,A134,A141,A142,A143,A145,
A146,A147,A148,A149,ARYTM_1:28
.= +(*(x,y),*(x,z)) by A130,A133,A137,A138,A139,A147,ZFMISC_1:33;
suppose
A150:  +(y,z) = 0;
then A151:  y'' = z'' by A141,A142,ARYTM_1:10;
A152: xy' = x' *' y' by A133,A137,ZFMISC_1:33;
A153: x' = x'' by A128,A131,ZFMISC_1:33;
thus *(x,+(y,z)) = 0 by A150,Th14
.= +(*(x,y),*(x,z)) by A130,A132,A134,A138,A139,A140,A151,A152,A153,
ARYTM_1:18;
end;
hence thesis;
suppose
A154: not z'' <=' y'';
then A155: +(y,z) = [0,z'' -' y''] by A136,ARYTM_1:def 2;
then +(y,z) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider x''',yz' being Element of REAL+ such that
A156: x = [0,x'''] and
A157: +(y,z) = [0,yz'] and
A158: *(x,+(y,z)) = yz' *' x''' by A126,Def3;
A159: x' = x'' by A128,A131,ZFMISC_1:33;
A160: x'' = x''' by A128,A156,ZFMISC_1:33;
thus *(x,+(y,z)) = x''' *' (z'' -' y'') by A155,A157,A158,ZFMISC_1:33
.= (x'' *' z') - (x' *' y') by A132,A134,A140,A154,A159,A160,ARYTM_1:26
.= +(*(x,y),*(x,z)) by A130,A133,A137,A138,A139,ZFMISC_1:33;
end;
hence thesis;
suppose
A161: y = 0;
hence *(x,+(y,z)) = *(x,z) by Th13
.= +(o,*(x,z)) by Th13
.= +(*(x,y),*(x,z)) by A161,Th14;
end;
hence thesis;
suppose that
A162: x in RR and
A163: z in REAL+ and
A164: y in RR;
A165: x in [:{0},REAL+:] by A162,XBOOLE_0:def 4;
A166: y in [:{0},REAL+:] by A164,XBOOLE_0:def 4;
then consider x'',y' being Element of REAL+ such that
A167: x = [0,x''] and
A168: y = [0,y'] and
A169: *(x,y) = y' *' x'' by A165,Def3;
now per cases;
suppose z <> 0;
then consider x',z' being Element of REAL+ such that
A170: x = [0,x'] and
A171: z = z' and
A172: *(x,z) = [0,z' *' x'] by A163,A165,Def3;
consider z'',y'' being Element of REAL+ such that
A173: z = z'' and
A174: y = [0,y''] and
A175: +(z,y) = z'' - y'' by A166,A171,Def2;
*(x,z) in [:{0},REAL+:] by A172,Lm1,ZFMISC_1:106;
then consider xz',xy' being Element of REAL+ such that
A176: *(x,z) = [0,xz'] and
A177: *(x,y) = xy' and
A178: +(*(x,z),*(x,y)) = xy' - xz' by A169,Def2;
A179: y' = y'' by A168,A174,ZFMISC_1:33;
now per cases;
suppose
A180: y'' <=' z'';
then A181: +(z,y) = z'' -' y'' by A175,ARYTM_1:def 2;
now per cases;
suppose
A182: +(z,y) <> 0;
then consider x''',zy' being Element of REAL+ such that
A183: x = [0,x'''] and
A184: +(z,y) = zy' and
A185: *(x,+(z,y)) = [0,zy' *' x'''] by A165,A181,Def3;
A186: x' = x'' by A167,A170,ZFMISC_1:33;
A187: x'' = x''' by A167,A183,ZFMISC_1:33;
A188: y' = y'' by A168,A174,ZFMISC_1:33;
not x in {[0,0]} by NUMBERS:def 1,XBOOLE_0:def 4;
then x''' <> 0 by A183,TARSKI:def 1;
hence *(x,+(z,y)) = (x' *' y') - (x' *' z') by A171,A173,A180,A181,A182,A184,
A185,A186,A187,A188,ARYTM_1:28
.= +(*(x,z),*(x,y)) by A169,A172,A176,A177,A178,A186,ZFMISC_1:33;
suppose
A189:  +(z,y) = 0;
then A190:  z'' = y'' by A180,A181,ARYTM_1:10;
A191: xz' = x' *' z' by A172,A176,ZFMISC_1:33;
A192: x' = x'' by A167,A170,ZFMISC_1:33;
thus *(x,+(z,y)) = 0 by A189,Th14
.= +(*(x,z),*(x,y)) by A169,A171,A173,A177,A178,A179,A190,A191,A192,ARYTM_1
:18;
end;
hence thesis;
suppose
A193: not y'' <=' z'';
then A194: +(z,y) = [0,y'' -' z''] by A175,ARYTM_1:def 2;
then +(z,y) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider x''',zy' being Element of REAL+ such that
A195: x = [0,x'''] and
A196: +(z,y) = [0,zy'] and
A197: *(x,+(z,y)) = zy' *' x''' by A165,Def3;
A198: x' = x'' by A167,A170,ZFMISC_1:33;
A199: x'' = x''' by A167,A195,ZFMISC_1:33;
thus *(x,+(y,z)) = x''' *' (y'' -' z'') by A194,A196,A197,ZFMISC_1:33
.= (x'' *' y') - (x' *' z') by A171,A173,A179,A193,A198,A199,ARYTM_1:26
.= +(*(x,y),*(x,z)) by A169,A172,A176,A177,A178,ZFMISC_1:33;
end;
hence thesis;
suppose
A200: z = 0;
hence *(x,+(y,z)) = *(x,y) by Th13
.= +(*(x,y),o) by Th13
.= +(*(x,y),*(x,z)) by A200,Th14;
end;
hence thesis;
suppose that
A201: x in RR and
A202: y in RR and
A203: z in RR;
A204: x in [:{0},REAL+:] by A201,XBOOLE_0:def 4;
A205: y in [:{0},REAL+:] by A202,XBOOLE_0:def 4;
A206: z in [:{0},REAL+:] by A203,XBOOLE_0:def 4;
consider x',y' being Element of REAL+ such that
A207: x = [0,x'] and
A208: y = [0,y'] and
A209: *(x,y) = y' *' x' by A204,A205,Def3;
consider x'',z' being Element of REAL+ such that
A210: x = [0,x''] and
A211: z = [0,z'] and
A212: *(x,z) = z' *' x'' by A204,A206,Def3;
consider xy',xz' being Element of REAL+ such that
A213: *(x,y) = xy' and
A214: *(x,z) = xz' and
A215: +(*(x,y),*(x,z)) = xy' + xz' by A209,A212,Def2;
not(y in REAL+ & z in REAL+) &
not(y in REAL+ & z in [:{0},REAL+:]) &
not(y in [:{0},REAL+:] & z in REAL+) by A205,A206,Th5,XBOOLE_0:3;
then consider y'',z'' being Element of REAL+ such that
A216: y = [0,y''] and
A217: z = [0,z''] and
A218: +(y,z) = [0,y'' + z''] by Def2;
+(y,z) in [:{0},REAL+:] by A218,Lm1,ZFMISC_1:106;
then consider x''',yz' being Element of REAL+ such that
A219: x = [0,x'''] and
A220: +(y,z) = [0,yz'] and
A221: *(x,+(y,z)) = yz' *' x''' by A204,Def3;
A222: x' = x'' by A207,A210,ZFMISC_1:33;
A223: x' = x''' by A207,A219,ZFMISC_1:33;
A224: y' = y'' by A208,A216,ZFMISC_1:33;
A225: z' = z'' by A211,A217,ZFMISC_1:33;
thus *(x,+(y,z)) = x''' *' (y'' + z'') by A218,A220,A221,ZFMISC_1:33
.= +(*(x,y),*(x,z)) by A209,A212,A213,A214,A215,A222,A223,A224,A225,
ARYTM_2:14;
suppose
A226: not(x in REAL+ & y in REAL+ & z in REAL+) &
not(x in REAL+ & y in REAL+ & z in RR) &
not(x in REAL+ & y in RR & z in REAL+) &
not(x in REAL+ & y in RR & z in RR) &
not(x in RR & y in REAL+ & z in REAL+) &
not(x in RR & y in REAL+ & z in RR) &
not(x in RR & y in RR & z in REAL+) &
not(x in RR & y in RR & z in RR);
A227: not [0,0] in REAL+ by ARYTM_2:3;
REAL = (REAL+ \ {[0,0]}) \/ ([:{0},REAL+:] \ {[0,0]})
by NUMBERS:def 1,XBOOLE_1:42
.= REAL+ \/ RR by A227,ZFMISC_1:65;
hence *(x,+(y,z)) = +(*(x,y),*(x,z)) by A226,XBOOLE_0:def 2;
end;

theorem
for x,y being Element of REAL holds *(opp x,y) = opp *(x,y)
proof let x,y be Element of REAL;
+(*(opp x,y),*(x,y)) = *(+(opp x,x), y) by Th16
.= *(o,y) by Def4
.= 0 by Th14;
hence *(opp x,y) = opp *(x,y) by Def4;
end;

theorem Th18:
for x being Element of REAL holds *(x,x) in REAL+
proof let x be Element of REAL;
A1: x in REAL+ \/ [:{{}},REAL+:] by NUMBERS:def 1,XBOOLE_0:def 4;
per cases by A1,XBOOLE_0:def 2;
suppose x in REAL+;
then ex x',y' being Element of REAL+ st x = x' & x = y' & *(x,x) = x' *' y'
by Def3;
hence *(x,x) in REAL+;
suppose x in [:{0},REAL+:];
then ex x',y' being Element of REAL+ st x = [0,x'] & x = [0,y'] &
*(x,x) = y' *' x' by Def3;
hence *(x,x) in REAL+;
end;

theorem
for x,y st +(*(x,x),*(y,y)) = 0 holds x = 0
proof let x,y such that
A1: +(*(x,x),*(y,y)) = 0;
*(x,x) in REAL+ & *(y,y) in REAL+ by Th18;
then consider x',y' being Element of REAL+ such that
A2:  *(x,x) = x' & *(y,y) = y' and
A3:  0 = x' + y' by A1,Def2;
A4:  x' = 0 by A3,ARYTM_2:6;
A5: x in REAL+ \/ [:{{}},REAL+:] by NUMBERS:def 1,XBOOLE_0:def 4;
per cases by A5,XBOOLE_0:def 2;
suppose x in REAL+;
then ex x',y' being Element of REAL+ st x = x' & x = y' & 0 = x' *' y'
by A2,A4,Def3;
hence x = 0 by ARYTM_1:2;
suppose x in [:{0},REAL+:];
then consider x',y' being Element of REAL+ such that
A6:  x = [0,x'] & x = [0,y'] and
A7:  0 = y' *' x' by A2,A4,Def3;
x' = y' by A6,ZFMISC_1:33;
then x' = 0 by A7,ARYTM_1:2;
then x in {[0,0]} by A6,TARSKI:def 1;
hence x = 0 by NUMBERS:def 1,XBOOLE_0:def 4;
end;

theorem Th20:
for x,y,z being Element of REAL st x <> 0 & *(x,y) = one & *(x,z) = one
holds y = z
proof let x,y,z be Element of REAL;
assume that
A1:    x <> 0 and
A2:    *(x,y) = one and
A3:    *(x,z) = one;
A4:     for x,y being Element of REAL st *(x,y) =one holds x <> 0
proof let x,y be Element of REAL such that
A5:       *(x,y) = one and
A6:       x = 0;
A7:       not x in [:{0},REAL+:] by A6,Th5,ARYTM_2:21,XBOOLE_0:3;
A8:      y in REAL+ \/ [:{0},REAL+:] by NUMBERS:def 1,XBOOLE_0:def 4;
per cases by A8,XBOOLE_0:def 2;
suppose y in REAL+;
then ex x',y' being Element of REAL+
st x = x' & y = y' & one = x' *' y'
by A5,A6,Def3,ARYTM_2:21;
suppose y in [:{0},REAL+:];
then not y in REAL+ by Th5,XBOOLE_0:3;
end;
then A9:    y <> 0 by A2;
A10:    z <> 0 by A3,A4;
per cases;
suppose x in REAL+ & y in REAL+ & z in REAL+;
then (ex x',y' being Element of REAL+ st x = x' & y = y' & one = x' *'
y')
& ex x',y' being Element of REAL+ st x = x' & z = y' & one = x' *' y'
by A2,A3,Def3;
hence y = z by A1,Th8;
suppose that
A11:    x in [:{0},REAL+:] and
A12:    y in [:{0},REAL+:] and
A13:    z in [:{0},REAL+:];
consider x',y' being Element of REAL+ such that
A14:     x = [0,x'] and
A15:     y = [0,y'] and
A16:     one = y' *' x' by A2,A11,A12,Def3;
consider x'',z' being Element of REAL+ such that
A17:     x = [0,x''] and
A18:     z = [0,z'] and
A19:     one = z' *' x'' by A3,A11,A13,Def3;
A20:     x' = x'' by A14,A17,ZFMISC_1:33;
x' <> 0 by A14,Th3;
hence y = z by A15,A16,A18,A19,A20,Th8;
suppose x in REAL+ & y in [:{0},REAL+:];
then ex x',y' being Element of REAL+ st
x = x' & y = [0,y'] & one = [0,x' *' y'] by A1,A2,Def3;
hence y = z by Lm2;
suppose x in [:{0},REAL+:] & y in REAL+;
then ex x',y' being Element of REAL+ st
x = [0,x'] & y = y' & one = [0,y' *' x'] by A2,A9,Def3;
hence y = z by Lm2;
suppose x in [:{0},REAL+:] & z in REAL+;
then ex x',z' being Element of REAL+ st
x = [0,x'] & z = z' & one = [0,z' *' x'] by A3,A10,Def3;
hence y = z by Lm2;
suppose x in REAL+ & z in [:{0},REAL+:];
then ex x',z' being Element of REAL+ st
x = x' & z = [0,z'] & one = [0,x' *' z'] by A1,A3,Def3;
hence y = z by Lm2;
suppose not (x in REAL+ & y in REAL+) &
not (x in REAL+ & y in [:{0},REAL+:] & x <> 0) &
not (y in REAL+ & x in [:{0},REAL+:] & y <> 0) &
not (x in [:{0},REAL+:] & y in [:{0},REAL+:]);
hence thesis by A2,Def3;
suppose not (x in REAL+ & z in REAL+) &
not (x in REAL+ & z in [:{0},REAL+:] & x <> 0) &
not (z in REAL+ & x in [:{0},REAL+:] & z <> 0) &
not (x in [:{0},REAL+:] & z in [:{0},REAL+:]);
hence thesis by A3,Def3;
end;

theorem Th21:
for x,y st y = one holds *(x,y) = x
proof let x,y such that
A1: y = one;
per cases;
suppose x = 0;
hence thesis by Th14;
suppose
A2:  x <> 0;
A3: now assume
A4:    inv x = 0;
thus one = *(x, inv x) by A2,Def5
.= 0 by A4,Th14;
end;
A5: *(x,inv x) = one by A2,Def5;
A6: ex x',y' being Element of REAL+ st y = x' & y = y' & *(y,y) = x' *' y'
by A1,Def3,ARYTM_2:21;
*(*(x,y), inv x) = *(*(x,inv x), y) by Th15
.= *(y,y) by A1,A2,Def5
.= one by A1,A6,ARYTM_2:16;
hence *(x,y) = x by A3,A5,Th20;
end;

reconsider j = one as Element of REAL by Th1,ARYTM_2:21;

theorem
for x,y st y <> 0 holds *(*(x,y),inv y) = x
proof let x,y such that
A1: y <> 0;
thus *(*(x,y),inv y) = *(x,*(y,inv y)) by Th15
.= *(x,j) by A1,Def5
.= x by Th21;
end;

theorem Th23:
for x,y st *(x,y) = 0 holds x = 0 or y = 0
proof let x,y such that
A1: *(x,y) = 0 and
A2: x <> 0;
A3:  *(x, inv x) = one by A2,Def5;
thus y = *(j,y) by Th21
.= *(*(x,y),inv x) by A3,Th15
.= 0 by A1,Th14;
end;

theorem
for x,y holds inv *(x,y) = *(inv x, inv y)
proof let x,y;
per cases;
suppose
A1:  x = 0 or y = 0;
then A2:  *(x,y) = 0 by Th14;
A3: inv x = 0 or inv y = 0 by A1,Def5;
thus inv *(x,y) = 0 by A2,Def5
.= *(inv x, inv y) by A3,Th14;
suppose that
A4: x <> 0 and
A5: y <> 0;
A6: *(y,inv y) = one by A5,Def5;
A7: *(x,inv x) = one by A4,Def5;
A8:  *(x,y) <> 0 by A4,A5,Th23;
*(*(x,y),*(inv x, inv y)) = *(*(*(x,y),inv x), inv y) by Th15
.= *(*(j,y), inv y) by A7,Th15
.= one by A6,Th21;
hence inv *(x,y) = *(inv x, inv y) by A8,Def5;
end;

theorem Th25:
for x,y,z being Element of REAL holds +(x,+(y,z)) = +(+(x,y),z)
proof let x,y,z be Element of REAL;
A1: x in REAL+ \/ [:{0},REAL+:] & y in REAL+ \/ [:{0},REAL+:] &
z in REAL+ \/ [:{0},REAL+:] by NUMBERS:def 1,XBOOLE_0:def 4;
per cases by A1,XBOOLE_0:def 2;
suppose that
A2: x in REAL+ and
A3: y in REAL+ and
A4: z in REAL+;
consider y',z' being Element of REAL+ such that
A5: y = y' and
A6: z = z' and
A7: +(y,z) = y' + z' by A3,A4,Def2;
consider x',yz' being Element of REAL+ such that
A8: x = x' and
A9: +(y,z) = yz' and
A10: +(x,+(y,z)) = x' + yz' by A2,A7,Def2;
consider x'',y'' being Element of REAL+ such that
A11: x = x'' and
A12: y = y'' and
A13: +(x,y) = x'' + y'' by A2,A3,Def2;
consider xy'',z'' being Element of REAL+ such that
A14: +(x,y) = xy'' and
A15: z = z'' and
A16: +(+(x,y),z) = xy'' + z'' by A4,A13,Def2;
thus +(x,+(y,z)) = +(+(x,y),z) by A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A15,A16,
ARYTM_2:7;
suppose that
A17: x in REAL+ and
A18: y in REAL+ and
A19: z in [:{0},REAL+:];
consider y',z' being Element of REAL+ such that
A20: y = y' and
A21: z = [0,z'] and
A22: +(y,z) = y' - z' by A18,A19,Def2;
consider x'',y'' being Element of REAL+ such that
A23: x = x'' and
A24: y = y'' and
A25: +(x,y) = x'' + y'' by A17,A18,Def2;
consider xy'',z'' being Element of REAL+ such that
A26: +(x,y) = xy'' and
A27: z = [0,z''] and
A28: +(+(x,y),z) = xy'' - z'' by A19,A25,Def2;
A29:  z' = z'' by A21,A27,ZFMISC_1:33;
now per cases;
suppose
A30:   z' <=' y';
then A31:   +(y,z) = y' -' z' by A22,ARYTM_1:def 2;
then consider x',yz' being Element of REAL+ such that
A32:  x = x' and
A33:  +(y,z) = yz' and
A34:  +(x,+(y,z)) = x' + yz' by A17,Def2;
thus +(x,+(y,z)) = +(+(x,y),z) by A20,A23,A24,A25,A26,A28,A29,A30,A31,A32,
A33,A34,ARYTM_1:20;
suppose
A35:  not z' <=' y';
then A36:   +(y,z) = [0,z' -' y'] by A22,ARYTM_1:def 2;
then +(y,z) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider x',yz' being Element of REAL+ such that
A37:  x = x' and
A38:  +(y,z) = [0,yz'] and
A39:  +(x,+(y,z)) = x' - yz' by A17,Def2;
thus +(x,+(y,z)) = x' - (z' -' y') by A36,A38,A39,ZFMISC_1:33
.= +(+(x,y),z) by A20,A23,A24,A25,A26,A28,A29,A35,A37,ARYTM_1:21;
end;
hence thesis;
suppose that
A40: x in REAL+ and
A41: y in [:{0},REAL+:] and
A42: z in REAL+;
consider z',y' being Element of REAL+ such that
A43: z = z' and
A44: y = [0,y'] and
A45: +(y,z) = z' - y' by A41,A42,Def2;
consider x'',y'' being Element of REAL+ such that
A46: x = x'' and
A47: y = [0,y''] and
A48: +(x,y) = x'' - y'' by A40,A41,Def2;
A49: y' = y'' by A44,A47,ZFMISC_1:33;
now per cases;
suppose that
A50: y' <=' x'' and
A51: y' <=' z';
A52: +(y,z) = z' -' y' by A45,A51,ARYTM_1:def 2;
then consider x',yz' being Element of REAL+ such that
A53:  x = x' and
A54:  +(y,z) = yz' and
A55:  +(x,+(y,z)) = x' + yz' by A40,Def2;
A56: +(x,y) = x' -' y' by A46,A48,A49,A50,A53,ARYTM_1:def 2;
then consider z'',xy'' being Element of REAL+ such that
A57: z = z'' and
A58: +(x,y) = xy'' and
A59: +(z,+(x,y)) = z'' + xy'' by A42,Def2;
thus +(x,+(y,z)) = +(+(x,y),z) by A43,A46,A50,A51,A52,A53,A54,A55,A56,A57,
A58,A59,ARYTM_1:12;
suppose that
A60: y' <=' x'' and
A61: not y' <=' z';
A62:   +(y,z) = [0,y' -' z'] by A45,A61,ARYTM_1:def 2;
then +(y,z) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider x',yz' being Element of REAL+ such that
A63:  x = x' and
A64:  +(y,z) = [0,yz'] and
A65:  +(x,+(y,z)) = x' - yz' by A40,Def2;
A66: +(x,y) = x'' -' y' by A48,A49,A60,ARYTM_1:def 2;
then consider z'',xy'' being Element of REAL+ such that
A67: z = z'' and
A68: +(x,y) = xy'' and
A69: +(z,+(x,y)) = z'' + xy'' by A42,Def2;
thus +(x,+(y,z)) = x' - (y' -' z') by A62,A64,A65,ZFMISC_1:33
.= +(+(x,y),z) by A43,A46,A60,A61,A63,A66,A67,A68,A69,ARYTM_1:22;
suppose that
A70: not y' <=' x'' and
A71: y' <=' z';
A72:   +(y,x) = [0,y' -' x''] by A48,A49,A70,ARYTM_1:def 2;
then +(y,x) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider z'',yx'' being Element of REAL+ such that
A73:  z = z'' and
A74:  +(y,x) = [0,yx''] and
A75:  +(z,+(y,x)) = z'' - yx'' by A42,Def2;
A76: +(z,y) = z' -' y' by A45,A71,ARYTM_1:def 2;
then consider x',zy'' being Element of REAL+ such that
A77: x = x' and
A78: +(z,y) = zy'' and
A79: +(x,+(z,y)) = x' + zy'' by A40,Def2;
thus +(x,+(y,z)) = z'' - (y' -' x'') by A43,A46,A70,A71,A73,A76,A77,A78,A79,
ARYTM_1:22
.= +(+(x,y),z) by A72,A74,A75,ZFMISC_1:33;
suppose that
A80: not y' <=' x'' and
A81: not y' <=' z';
A82:   +(y,x) = [0,y' -' x''] by A48,A49,A80,ARYTM_1:def 2;
then +(y,x) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider z'',yx'' being Element of REAL+ such that
A83:  z = z'' and
A84:  +(y,x) = [0,yx''] and
A85:  +(z,+(y,x)) = z'' - yx'' by A42,Def2;
A86:   +(y,z) = [0,y' -' z'] by A45,A81,ARYTM_1:def 2;
then +(y,z) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider x',yz' being Element of REAL+ such that
A87:  x = x' and
A88:  +(y,z) = [0,yz'] and
A89:  +(x,+(y,z)) = x' - yz' by A40,Def2;
thus +(x,+(y,z)) = x' - (y' -' z') by A86,A88,A89,ZFMISC_1:33
.= z'' - (y' -' x'') by A43,A46,A80,A81,A83,A87,ARYTM_1:23
.= +(+(x,y),z) by A82,A84,A85,ZFMISC_1:33;
end;
hence thesis;
suppose that
A90: x in REAL+ and
A91: y in [:{0},REAL+:] and
A92: z in [:{0},REAL+:];
not(z in REAL+ & y in REAL+) &
not(z in REAL+ & y in [:{0},REAL+:]) &
not(y in REAL+ & z in [:{0},REAL+:]) by A91,A92,Th5,XBOOLE_0:3;
then consider y',z' being Element of REAL+ such that
A93: y = [0,y'] and
A94: z = [0,z'] and
A95: +(y,z) = [0,y' + z'] by Def2;
consider x'',y'' being Element of REAL+ such that
A96: x = x'' and
A97: y = [0,y''] and
A98: +(x,y) = x'' - y'' by A90,A91,Def2;
+(y,z) in [:{0},REAL+:] by A95,Lm1,ZFMISC_1:106;
then consider x',yz' being Element of REAL+ such that
A99: x = x' and
A100: +(y,z) = [0,yz'] and
A101: +(x,+(y,z)) = x' - yz' by A90,Def2;
A102:  y' = y'' by A93,A97,ZFMISC_1:33;
now per cases;
suppose
A103: y'' <=' x'';
then A104: +(x,y) = x'' -' y'' by A98,ARYTM_1:def 2;
then consider xy'',z'' being Element of REAL+ such that
A105: +(x,y) = xy'' and
A106: z = [0,z''] and
A107: +(+(x,y),z) = xy'' - z'' by A92,Def2;
A108:  z' = z'' by A94,A106,ZFMISC_1:33;
thus +(x,+(y,z)) = x' - (y' + z') by A95,A100,A101,ZFMISC_1:33
.= +(+(x,y),z) by A96,A99,A102,A103,A104,A105,A107,A108,ARYTM_1:24;
suppose
A109: not y'' <=' x'';
then A110:   +(x,y) = [0,y'' -' x''] by A98,ARYTM_1:def 2;
then +(x,y) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then not(z in REAL+ & +(x,y) in REAL+) &
not(z in REAL+ & +(x,y) in [:{0},REAL+:]) &
not(+(x,y) in REAL+ & z in [:{0},REAL+:]) by A92,Th5,XBOOLE_0:3;
then consider xy'',z'' being Element of REAL+ such that
A111:  +(x,y) = [0,xy''] and
A112:  z = [0,z''] and
A113:  +(+(x,y),z) = [0,xy'' + z''] by Def2;
A114:  z' = z'' by A94,A112,ZFMISC_1:33;
A115:  yz' = z' + y' by A95,A100,ZFMISC_1:33;
then y'' <=' yz' by A102,ARYTM_2:20;
then not yz' <=' x' by A96,A99,A109,ARYTM_1:3;
hence +(x,+(y,z)) = [0,z' + y' -' x'] by A101,A115,ARYTM_1:def 2
.= [0,z'' + (y'' -' x'')] by A96,A99,A102,A109,A114,ARYTM_1:13
.= +(+(x,y),z) by A110,A111,A113,ZFMISC_1:33;
end;
hence thesis;
suppose that
A116: z in REAL+ and
A117: y in REAL+ and
A118: x in [:{0},REAL+:];
consider y',x' being Element of REAL+ such that
A119: y = y' and
A120: x = [0,x'] and
A121: +(y,x) = y' - x' by A117,A118,Def2;
consider z'',y'' being Element of REAL+ such that
A122: z = z'' and
A123: y = y'' and
A124: +(z,y) = z'' + y'' by A116,A117,Def2;
consider zy'',x'' being Element of REAL+ such that
A125: +(z,y) = zy'' and
A126: x = [0,x''] and
A127: +(+(z,y),x) = zy'' - x'' by A118,A124,Def2;
A128:  x' = x'' by A120,A126,ZFMISC_1:33;
now per cases;
suppose
A129:   x' <=' y';
then A130:   +(y,x) = y' -' x' by A121,ARYTM_1:def 2;
then consider z',yx' being Element of REAL+ such that
A131:  z = z' and
A132:  +(y,x) = yx' and
A133:  +(z,+(y,x)) = z' + yx' by A116,Def2;
thus +(z,+(y,x)) = +(+(z,y),x) by A119,A122,A123,A124,A125,A127,A128,A129,
A130,A131,A132,A133,ARYTM_1:20;
suppose
A134:  not x' <=' y';
then A135:   +(y,x) = [0,x' -' y'] by A121,ARYTM_1:def 2;
then +(y,x) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then consider z',yx' being Element of REAL+ such that
A136:  z = z' and
A137:  +(y,x) = [0,yx'] and
A138:  +(z,+(y,x)) = z' - yx' by A116,Def2;
thus +(z,+(y,x)) = z' - (x' -' y') by A135,A137,A138,ZFMISC_1:33
.= +(+(z,y),x) by A119,A122,A123,A124,A125,A127,A128,A134,A136,
ARYTM_1:21;
end;
hence thesis;
suppose that
A139: x in [:{0},REAL+:] and
A140: y in REAL+ and
A141: z in [:{0},REAL+:];
consider y',z' being Element of REAL+ such that
A142: y = y' and
A143: z = [0,z'] and
A144: +(y,z) = y' - z' by A140,A141,Def2;
consider x'',y'' being Element of REAL+ such that
A145: x = [0,x''] and
A146: y = y'' and
A147: +(x,y) = y'' - x'' by A139,A140,Def2;
now per cases;
suppose that
A148: x'' <=' y'' and
A149: z' <=' y';
A150: +(y,z) = y' -' z' by A144,A149,ARYTM_1:def 2;
then consider x',yz' being Element of REAL+ such that
A151:  x = [0,x'] and
A152:  +(y,z) = yz' and
A153:  +(x,+(y,z)) = yz' - x' by A139,Def2;
A154:  x' = x'' by A145,A151,ZFMISC_1:33;
then A155: +(x,y) = y' -' x' by A142,A146,A147,A148,ARYTM_1:def 2;
then consider z'',xy'' being Element of REAL+ such that
A156: z = [0,z''] and
A157: +(x,y) = xy'' and
A158: +(z,+(x,y)) = xy'' - z'' by A141,Def2;
z' = z'' by A143,A156,ZFMISC_1:33;
hence +(x,+(y,z)) = +(+(x,y),z) by A142,A146,A148,A149,A150,A152,A153,A154,
A155,A157,A158,ARYTM_1:25;
suppose that
A159: not x'' <=' y'' and
A160: z' <=' y';
A161: +(y,z) = y' -' z' by A144,A160,ARYTM_1:def 2;
then consider x',yz' being Element of REAL+ such that
A162:  x = [0,x'] and
A163:  +(y,z) = yz' and
A164:  +(x,+(y,z)) = yz' - x' by A139,Def2;
A165:  x' = x'' by A145,A162,ZFMISC_1:33;
A166:   +(y,x) = [0,x'' -' y''] by A147,A159,ARYTM_1:def 2;
then +(y,x) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then not(z in REAL+ & +(x,y) in REAL+) &
not(z in REAL+ & +(x,y) in [:{0},REAL+:]) &
not(+(x,y) in REAL+ & z in [:{0},REAL+:]) by A141,Th5,XBOOLE_0:3;
then consider z'',yx' being Element of REAL+ such that
A167:  z = [0,z''] and
A168:  +(y,x) = [0,yx'] and
A169:  +(z,+(y,x)) = [0,z'' + yx'] by Def2;
A170:  z' = z'' by A143,A167,ZFMISC_1:33;
yz' <=' y' by A161,A163,ARYTM_1:11;
then not x' <=' yz' by A142,A146,A159,A165,ARYTM_1:3;
hence +(x,+(y,z)) = [0,x' -' (y' -' z')] by A161,A163,A164,ARYTM_1:def 2
.= [0,x'' -' y'' + z''] by A142,A146,A159,A160,A165,A170,ARYTM_1:14
.= +(+(x,y),z) by A166,A168,A169,ZFMISC_1:33;
suppose that
A171: not z' <=' y' and
A172: x'' <=' y'';
A173: +(y,x) = y'' -' x'' by A147,A172,ARYTM_1:def 2;
then consider z'',yx'' being Element of REAL+ such that
A174:  z = [0,z''] and
A175:  +(y,x) = yx'' and
A176:  +(z,+(y,x)) = yx'' - z'' by A141,Def2;
A177:  z'' = z' by A143,A174,ZFMISC_1:33;
A178:   +(y,z) = [0,z' -' y'] by A144,A171,ARYTM_1:def 2;
then +(y,z) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then not(x in REAL+ & +(z,y) in REAL+) &
not(x in REAL+ & +(z,y) in [:{0},REAL+:]) &
not(+(z,y) in REAL+ & x in [:{0},REAL+:]) by A139,Th5,XBOOLE_0:3;
then consider x',yz'' being Element of REAL+ such that
A179:  x = [0,x'] and
A180:  +(y,z) = [0,yz''] and
A181:  +(x,+(y,z)) = [0,x' + yz''] by Def2;
A182:  x'' = x' by A145,A179,ZFMISC_1:33;
yx'' <=' y'' by A173,A175,ARYTM_1:11;
then A183:  not z'' <=' yx'' by A142,A146,A171,A177,ARYTM_1:3;
thus +(x,+(y,z)) = [0,z' -' y' + x'] by A178,A180,A181,ZFMISC_1:33
.= [0,z'' -' (y'' -' x'')] by A142,A146,A171,A172,A177,A182,ARYTM_1:
14
.= +(+(x,y),z) by A173,A175,A176,A183,ARYTM_1:def 2;
suppose that
A184: not x'' <=' y'' and
A185: not z' <=' y';
A186: +(y,z) = [0,z' -' y'] by A144,A185,ARYTM_1:def 2;
then +(y,z) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then not(x in REAL+ & +(z,y) in REAL+) &
not(x in REAL+ & +(z,y) in [:{0},REAL+:]) &
not(+(z,y) in REAL+ & x in [:{0},REAL+:]) by A139,Th5,XBOOLE_0:3;
then consider x',yz'' being Element of REAL+ such that
A187:  x = [0,x'] and
A188:  +(y,z) = [0,yz''] and
A189:  +(x,+(y,z)) = [0,x' + yz''] by Def2;
A190:   +(y,x) = [0,x'' -' y''] by A147,A184,ARYTM_1:def 2;
then +(y,x) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then not(z in REAL+ & +(x,y) in REAL+) &
not(z in REAL+ & +(x,y) in [:{0},REAL+:]) &
not(+(x,y) in REAL+ & z in [:{0},REAL+:]) by A141,Th5,XBOOLE_0:3;
then consider z'',yx' being Element of REAL+ such that
A191:  z = [0,z''] and
A192:  +(y,x) = [0,yx'] and
A193:  +(z,+(y,x)) = [0,z'' + yx'] by Def2;
A194:  z' = z'' by A143,A191,ZFMISC_1:33;
A195:  x' = x'' by A145,A187,ZFMISC_1:33;
thus +(x,+(y,z)) = [0,z' -' y' + x'] by A186,A188,A189,ZFMISC_1:33
.= [0,x'' -' y'' + z''] by A142,A146,A184,A185,A194,A195,ARYTM_1:15
.= +(+(x,y),z) by A190,A192,A193,ZFMISC_1:33;
end;
hence thesis;
suppose that
A196: z in REAL+ and
A197: y in [:{0},REAL+:] and
A198: x in [:{0},REAL+:];
not(x in REAL+ & y in REAL+) &
not(x in REAL+ & y in [:{0},REAL+:]) &
not(y in REAL+ & x in [:{0},REAL+:]) by A197,A198,Th5,XBOOLE_0:3;
then consider y',x' being Element of REAL+ such that
A199: y = [0,y'] and
A200: x = [0,x'] and
A201: +(y,x) = [0,y' + x'] by Def2;
consider z'',y'' being Element of REAL+ such that
A202: z = z'' and
A203: y = [0,y''] and
A204: +(z,y) = z'' - y'' by A196,A197,Def2;
+(y,x) in [:{0},REAL+:] by A201,Lm1,ZFMISC_1:106;
then consider z',yx' being Element of REAL+ such that
A205: z = z' and
A206: +(y,x) = [0,yx'] and
A207: +(z,+(y,x)) = z' - yx' by A196,Def2;
A208:  y' = y'' by A199,A203,ZFMISC_1:33;
now per cases;
suppose
A209: y'' <=' z'';
then A210: +(z,y) = z'' -' y'' by A204,ARYTM_1:def 2;
then consider zy'',x'' being Element of REAL+ such that
A211: +(z,y) = zy'' and
A212: x = [0,x''] and
A213: +(+(z,y),x) = zy'' - x'' by A198,Def2;
A214:  x' = x'' by A200,A212,ZFMISC_1:33;
thus +(z,+(y,x)) = z' - (y' + x') by A201,A206,A207,ZFMISC_1:33
.= +(+(z,y),x) by A202,A205,A208,A209,A210,A211,A213,A214,ARYTM_1:24;
suppose
A215: not y'' <=' z'';
then A216:   +(z,y) = [0,y'' -' z''] by A204,ARYTM_1:def 2;
then +(z,y) in [:{0},REAL+:] by Lm1,ZFMISC_1:106;
then not(x in REAL+ & +(z,y) in REAL+) &
not(x in REAL+ & +(z,y) in [:{0},REAL+:]) &
not(+(z,y) in REAL+ & x in [:{0},REAL+:]) by A198,Th5,XBOOLE_0:3;
then consider zy'',x'' being Element of REAL+ such that
A217:  +(z,y) = [0,zy''] and
A218:  x = [0,x''] and
A219:  +(+(z,y),x) = [0,zy'' + x''] by Def2;
A220:  x' = x'' by A200,A218,ZFMISC_1:33;
A221:  yx' = x' + y' by A201,A206,ZFMISC_1:33;
then y'' <=' yx' by A208,ARYTM_2:20;
then not yx' <=' z' by A202,A205,A215,ARYTM_1:3;
hence +(z,+(y,x)) = [0,x' + y' -' z'] by A207,A221,ARYTM_1:def 2
.= [0,x'' + (y'' -' z'')] by A202,A205,A208,A215,A220,ARYTM_1:13
.= +(+(z,y),x) by A216,A217,A219,ZFMISC_1:33;
end;
hence thesis;
suppose that
A222: x in [:{0},REAL+:] and
A223: y in [:{0},REAL+:] and
A224: z in [:{0},REAL+:];
not(z in REAL+ & y in REAL+) &
not(z in REAL+ & y in [:{0},REAL+:]) &
not(y in REAL+ & z in [:{0},REAL+:]) by A223,A224,Th5,XBOOLE_0:3;
then consider y',z' being Element of REAL+ such that
A225: y = [0,y'] and
A226: z = [0,z'] and
A227: +(y,z) = [0,y' + z'] by Def2;
not(x in REAL+ & y in REAL+) &
not(x in REAL+ & y in [:{0},REAL+:]) &
not(y in REAL+ & x in [:{0},REAL+:]) by A222,A223,Th5,XBOOLE_0:3;
then consider x'',y'' being Element of REAL+ such that
A228: x = [0,x''] and
A229: y = [0,y''] and
A230: +(x,y) = [0,x'' + y''] by Def2;
+(z,y) in [:{0},REAL+:] by A227,Lm1,ZFMISC_1:106;
then not(x in REAL+ & +(z,y) in REAL+) &
not(x in REAL+ & +(z,y) in [:{0},REAL+:]) &
not(+(z,y) in REAL+ & x in [:{0},REAL+:]) by A222,Th5,XBOOLE_0:3;
then consider x',yz' being Element of REAL+ such that
A231: x = [0,x'] and
A232: +(y,z) = [0,yz'] and
A233: +(x,+(y,z)) = [0,x' + yz'] by Def2;
+(x,y) in [:{0},REAL+:] by A230,Lm1,ZFMISC_1:106;
then not(z in REAL+ & +(x,y) in REAL+) &
not(z in REAL+ & +(x,y) in [:{0},REAL+:]) &
not(+(x,y) in REAL+ & z in [:{0},REAL+:]) by A224,Th5,XBOOLE_0:3;
then consider xy'',z'' being Element of REAL+ such that
A234: +(x,y) = [0,xy''] and
A235: z = [0,z''] and
A236: +(+(x,y),z) = [0,xy'' + z''] by Def2;
A237: z' = z'' by A226,A235,ZFMISC_1:33;
A238: x' = x'' by A228,A231,ZFMISC_1:33;
A239: y' = y'' by A225,A229,ZFMISC_1:33;
thus +(x,+(y,z)) = [0,z' + y' + x'] by A227,A232,A233,ZFMISC_1:33
.= [0,z'' + (y'' + x'')] by A237,A238,A239,ARYTM_2:7
.= +(+(x,y),z) by A230,A234,A236,ZFMISC_1:33;
end;

theorem
[*x,y*] in REAL implies y = 0
proof assume
A1: [*x,y*] in REAL;
assume y <> 0;
then [*x,y*] = (0,one) --> (x,y) by Def7;
end;

theorem
for x,y being Element of REAL holds opp +(x,y) = +(opp x, opp y)
proof let x,y be Element of REAL;
+(+(x,y),+(opp x, opp y))
= +(x,+(y,+(opp x, opp y))) by Th25
.= +(x,+(opp x,+(y, opp y))) by Th25
.= +(x,+(opp x,o)) by Def4
.= +(x,opp x) by Th13
.= 0 by Def4;
hence thesis by Def4;
end;

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