:: Basic Properties of Real Numbers
:: by Krzysztof Hryniewiecki
::
:: Received January 8, 1989
:: Copyright (c) 1990 Association of Mizar Users

:: REAL_1 semantic presentation

registration
cluster -> real Element of REAL ;
coherence
for b1 being Element of REAL holds b1 is real
proof end;
end;

definition
mode Real is Element of REAL ;
end;

registration
cluster positive Element of REAL ;
existence
ex b1 being Real st b1 is positive
proof end;
end;

definition
let x be Real;
:: original: -
redefine func - x -> Real;
coherence
- x is Real
by XREAL_0:def 1;
:: original: "
redefine func x " -> Real;
coherence
x " is Real
by XREAL_0:def 1;
end;

definition
let x be real number ;
let y be Real;
:: original: +
redefine func x + y -> Real;
coherence
x + y is Real
by XREAL_0:def 1;
:: original: *
redefine func x * y -> Real;
coherence
x * y is Real
by XREAL_0:def 1;
:: original: -
redefine func x - y -> Real;
coherence
x - y is Real
by XREAL_0:def 1;
:: original: /
redefine func x / y -> Real;
coherence
x / y is Real
by XREAL_0:def 1;
end;

definition
let x be Real;
let y be real number ;
:: original: +
redefine func x + y -> Real;
coherence
x + y is Real
by XREAL_0:def 1;
:: original: *
redefine func x * y -> Real;
coherence
x * y is Real
by XREAL_0:def 1;
:: original: -
redefine func x - y -> Real;
coherence
x - y is Real
by XREAL_0:def 1;
:: original: /
redefine func x / y -> Real;
coherence
x / y is Real
by XREAL_0:def 1;
end;

theorem :: REAL_1:1
canceled;

theorem :: REAL_1:2
canceled;

theorem :: REAL_1:3
canceled;

theorem :: REAL_1:4
canceled;

theorem :: REAL_1:5
canceled;

theorem :: REAL_1:6
canceled;

theorem :: REAL_1:7
canceled;

theorem :: REAL_1:8
canceled;

theorem :: REAL_1:9
canceled;

theorem :: REAL_1:10
canceled;

theorem :: REAL_1:11
canceled;

theorem :: REAL_1:12
canceled;

theorem :: REAL_1:13
canceled;

theorem :: REAL_1:14
canceled;

theorem :: REAL_1:15
canceled;

theorem :: REAL_1:16
canceled;

theorem :: REAL_1:17
canceled;

theorem :: REAL_1:18
canceled;

theorem :: REAL_1:19
canceled;

theorem :: REAL_1:20
canceled;

theorem :: REAL_1:21
canceled;

theorem :: REAL_1:22
canceled;

theorem :: REAL_1:23
canceled;

theorem :: REAL_1:24
canceled;

theorem :: REAL_1:25
for x being real number holds x - 0 = x ;

theorem :: REAL_1:26
- 0 = 0 ;

definition
let y, x be real number ;
canceled;
canceled;
canceled;
canceled;
redefine pred y <= x means :: REAL_1:def 5
( not x <= y or not x <> y );
compatibility
( not x < y iff ( not x <= y or not x <> y ) )
by XXREAL_0:1;
end;

:: deftheorem REAL_1:def 1 :
canceled;

:: deftheorem REAL_1:def 2 :
canceled;

:: deftheorem REAL_1:def 3 :
canceled;

:: deftheorem REAL_1:def 4 :
canceled;

:: deftheorem defines < REAL_1:def 5 :
for y, x being real number holds
( not x < y iff ( not x <= y or not x <> y ) );

theorem :: REAL_1:27
canceled;

theorem :: REAL_1:28
canceled;

theorem :: REAL_1:29
canceled;

theorem :: REAL_1:30
canceled;

theorem :: REAL_1:31
canceled;

theorem :: REAL_1:32
canceled;

theorem :: REAL_1:33
canceled;

theorem :: REAL_1:34
canceled;

theorem :: REAL_1:35
canceled;

theorem :: REAL_1:36
canceled;

theorem :: REAL_1:37
canceled;

theorem :: REAL_1:38
canceled;

theorem :: REAL_1:39
canceled;

theorem :: REAL_1:40
canceled;

theorem :: REAL_1:41
canceled;

theorem :: REAL_1:42
canceled;

theorem :: REAL_1:43
canceled;

theorem :: REAL_1:44
canceled;

theorem :: REAL_1:45
canceled;

theorem :: REAL_1:46
canceled;

theorem :: REAL_1:47
canceled;

theorem :: REAL_1:48
canceled;

theorem :: REAL_1:49
canceled;

theorem :: REAL_1:50
canceled;

theorem :: REAL_1:51
canceled;

theorem :: REAL_1:52
canceled;

theorem :: REAL_1:53
canceled;

theorem :: REAL_1:54
canceled;

theorem :: REAL_1:55
canceled;

theorem :: REAL_1:56
canceled;

theorem :: REAL_1:57
canceled;

theorem :: REAL_1:58
canceled;

theorem :: REAL_1:59
canceled;

theorem :: REAL_1:60
canceled;

theorem :: REAL_1:61
canceled;

theorem :: REAL_1:62
canceled;

theorem :: REAL_1:63
canceled;

theorem :: REAL_1:64
canceled;

theorem :: REAL_1:65
canceled;

theorem :: REAL_1:66
canceled;

theorem :: REAL_1:67
canceled;

theorem :: REAL_1:68
canceled;

theorem :: REAL_1:69
canceled;

theorem :: REAL_1:70
canceled;

theorem :: REAL_1:71
canceled;

theorem :: REAL_1:72
canceled;

theorem :: REAL_1:73
for z, x, y being real number st 0 < z holds
( x < y iff x / z < y / z ) by XREAL_1:74, XREAL_1:76;

theorem :: REAL_1:74
for z, x, y being real number st z < 0 holds
( x < y iff y / z < x / z ) by XREAL_1:75, XREAL_1:77;

scheme :: REAL_1:sch 1
SepReal{ P1[ Real] } :
ex X being Subset of REAL st
for x being Real holds
( x in X iff P1[x] )
proof end;

theorem :: REAL_1:75
canceled;

theorem :: REAL_1:76
canceled;

theorem :: REAL_1:77
canceled;

theorem :: REAL_1:78
canceled;

theorem :: REAL_1:79
canceled;

theorem :: REAL_1:80
canceled;

theorem :: REAL_1:81
canceled;

theorem :: REAL_1:82
canceled;

theorem :: REAL_1:83
canceled;

theorem :: REAL_1:84
canceled;

theorem :: REAL_1:85
canceled;

theorem :: REAL_1:86
canceled;

theorem :: REAL_1:87
canceled;

theorem :: REAL_1:88
canceled;

theorem :: REAL_1:89
canceled;

theorem :: REAL_1:90
canceled;

theorem :: REAL_1:91
canceled;

theorem :: REAL_1:92
for x, y, z, t being real number holds
( ( x <= y & z <= t implies x - t <= y - z ) & ( ( ( x < y & z <= t ) or ( x <= y & z < t ) ) implies x - t < y - z ) ) by XREAL_1:15, XREAL_1:16, XREAL_1:17;