:: XREAL_0 semantic presentation

definition
let r be number ;
attr a1 is real means :Def1: :: XREAL_0:def 1
r in REAL ;
end;

:: deftheorem Def1 defines real XREAL_0:def 1 :
for r being number holds
( r is real iff r in REAL );

registration
cluster -infty -> non real ;
coherence
not -infty is real
proof end;
cluster +infty -> non real ;
coherence
not +infty is real
proof end;
end;

registration
cluster natural -> real set ;
coherence
for b1 being number st b1 is natural holds
b1 is real
proof end;
cluster real -> complex set ;
coherence
for b1 being number st b1 is real holds
b1 is complex
proof end;
end;

registration
cluster complex real set ;
existence
ex b1 being number st b1 is real
proof end;
cluster real -> ext-real set ;
coherence
for b1 being number st b1 is real holds
b1 is ext-real
proof end;
end;

Lemma24: for x being real number
for x1, x2 being Element of REAL st x = [*x1,x2*] holds
( x2 = 0 & x = x1 )
proof end;

registration
let x be real number ;
cluster - a1 -> ext-real real ;
coherence
- x is real
proof end;
cluster a1 " -> ext-real real ;
coherence
x " is real
proof end;
let y be real number ;
cluster a1 + a2 -> ext-real real ;
coherence
x + y is real
proof end;
cluster a1 * a2 -> ext-real real ;
coherence
x * y is real
proof end;
end;

registration
let x be real number , y be real number ;
cluster a1 - a2 -> ext-real real ;
coherence
x - y is real
;
cluster a1 / a2 -> ext-real real ;
coherence
x / y is real
;
end;

Lemma47: for r, s being real number st r <= s holds
( ( r in REAL+ & s in REAL+ implies ex x', y' being Element of REAL+ st
( r = x' & s = y' & x' <=' y' ) ) & ( r in [:{0},REAL+ :] & s in [:{0},REAL+ :] implies ex x', y' being Element of REAL+ st
( r = [0,x'] & s = [0,y'] & y' <=' x' ) ) & ( ( not r in REAL+ or not s in REAL+ ) & ( not r in [:{0},REAL+ :] or not s in [:{0},REAL+ :] ) implies ( s in REAL+ & r in [:{0},REAL+ :] ) ) )
by XXREAL_0:def 5;

Lemma50: for r, s being real number st ( ( r in REAL+ & s in REAL+ & ex x', y' being Element of REAL+ st
( r = x' & s = y' & x' <=' y' ) ) or ( r in [:{0},REAL+ :] & s in [:{0},REAL+ :] & ex x', y' being Element of REAL+ st
( r = [0,x'] & s = [0,y'] & y' <=' x' ) ) or ( s in REAL+ & r in [:{0},REAL+ :] ) ) holds
r <= s
proof end;

Lemma51: {} in {{} }
by TARSKI:def 1;

Lemma52: for r, s being real number st r <= s & s <= r holds
r = s
proof end;

Lemma61: for r, s, t being real number st r <= s holds
r + t <= s + t
proof end;

Lemma143: for r, s, t being real number st r <= s & s <= t holds
r <= t
proof end;

reconsider z = 0 as Element of REAL+ by ARYTM_2:21;

Lemma145: not 0 in [:{0},REAL+ :]
by ARYTM_0:5, ARYTM_2:21, XBOOLE_0:3;

reconsider j = 1 as Element of REAL+ by ARYTM_2:21;

z <=' j
by ARYTM_1:6;

then Lemma147: 0 <= 1
by ;

1 + (- 1) = 0
;

then consider x1 being Element of REAL , x2 being Element of REAL , y1 being Element of REAL , y2 being Element of REAL such that
Lemma148: 1 = [*x1,x2*] and
Lemma149: - 1 = [*y1,y2*] and
Lemma150: 0 = [*(+ x1,y1),(+ x2,y2)*] by XCMPLX_0:def 4;

Lemma151: x1 = 1
by , ;

Lemma152: y1 = - 1
by , ;

Lemma153: + x1,y1 = 0
by , ;

E154: now
assume - 1 in REAL+ ;
then ex x', y' being Element of REAL+ st
( x1 = x' & y1 = y' & z = x' + y' ) by , , , ARYTM_0:def 2, ARYTM_2:21;
hence contradiction by , ARYTM_2:6;
end;

Lemma155: for r, s being real number st r >= 0 & s > 0 holds
r + s > 0
proof end;

Lemma156: for r, s being real number st r <= 0 & s < 0 holds
r + s < 0
proof end;

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

Lemma158: for r, s, t being real number st r <= s & 0 <= t holds
r * t <= s * t
proof end;

Lemma159: for r, s, t being real number holds (r * s) * t = r * (s * t)
;

Lemma160: for r, s being real number holds
( not r * s = 0 or r = 0 or s = 0 )
proof end;

Lemma161: for r, s being real number st r > 0 & s > 0 holds
r * s > 0
proof end;

Lemma162: for r, s being real number st r > 0 & s < 0 holds
r * s < 0
proof end;

Lemma163: for s, t being real number st s <= t holds
- t <= - s
proof end;

Lemma164: for r, s being real number st r <= 0 & s >= 0 holds
r * s <= 0
proof end;

registration
cluster complex ext-real positive real set ;
existence
ex b1 being real number st b1 is positive
proof end;
cluster complex ext-real negative real set ;
existence
ex b1 being real number st b1 is negative
proof end;
cluster zero complex ext-real real set ;
existence
ex b1 being real number st b1 is empty
proof end;
end;

registration
let r be non negative real number , s be non negative real number ;
cluster a1 + a2 -> ext-real non negative real ;
coherence
not r + s is negative
proof end;
end;

registration
let r be non positive real number , s be non positive real number ;
cluster a1 + a2 -> ext-real non positive real ;
coherence
not r + s is positive
proof end;
end;

registration
let r be positive real number ;
let s be non negative real number ;
cluster a1 + a2 -> ext-real positive non negative real ;
coherence
r + s is positive
proof end;
cluster a2 + a1 -> ext-real positive non negative real ;
coherence
s + r is positive
;
end;

registration
let r be negative real number ;
let s be non positive real number ;
cluster a1 + a2 -> ext-real non positive negative real ;
coherence
r + s is negative
proof end;
cluster a2 + a1 -> ext-real non positive negative real ;
coherence
s + r is negative
;
end;

registration
let r be non positive real number ;
cluster - a1 -> ext-real non negative real ;
coherence
not - r is negative
proof end;
end;

registration
let r be non negative real number ;
cluster - a1 -> ext-real non positive real ;
coherence
not - r is positive
proof end;
end;

registration
let r be non negative real number ;
let s be non positive real number ;
cluster a1 - a2 -> ext-real non negative real ;
coherence
not r - s is negative
;
cluster a2 - a1 -> ext-real non positive real ;
coherence
not s - r is positive
;
end;

registration
let r be positive real number ;
let s be non positive real number ;
cluster a1 - a2 -> ext-real positive non negative real ;
coherence
r - s is positive
;
cluster a2 - a1 -> ext-real non positive negative real ;
coherence
s - r is negative
;
end;

registration
let r be negative real number ;
let s be non negative real number ;
cluster a1 - a2 -> ext-real non positive negative real ;
coherence
r - s is negative
;
cluster a2 - a1 -> ext-real positive non negative real ;
coherence
s - r is positive
;
end;

registration
let r be non positive real number ;
let s be non negative real number ;
cluster a1 * a2 -> ext-real non positive real ;
coherence
not r * s is positive
proof end;
cluster a2 * a1 -> ext-real non positive real ;
coherence
not s * r is positive
;
end;

registration
let r be non positive real number , s be non positive real number ;
cluster a1 * a2 -> ext-real non negative real ;
coherence
not r * s is negative
proof end;
end;

registration
let r be non negative real number , s be non negative real number ;
cluster a1 * a2 -> ext-real non negative real ;
coherence
not r * s is negative
proof end;
end;

Lemma165: for r being real number st r " = 0 holds
r = 0
proof end;

registration
let r be non positive real number ;
cluster a1 " -> ext-real non positive real ;
coherence
not r " is positive
proof end;
end;

registration
let r be non negative real number ;
cluster a1 " -> ext-real non negative real ;
coherence
not r " is negative
proof end;
end;

registration
let r be non negative real number ;
let s be non positive real number ;
cluster a1 / a2 -> ext-real non positive real ;
coherence
not r / s is positive
;
cluster a2 / a1 -> ext-real non positive real ;
coherence
not s / r is positive
;
end;

registration
let r be non negative real number , s be non negative real number ;
cluster a1 / a2 -> ext-real non negative real ;
coherence
not r / s is negative
;
end;

registration
let r be non positive real number , s be non positive real number ;
cluster a1 / a2 -> ext-real non negative real ;
coherence
not r / s is negative
;
end;

registration
let r be real number ;
let s be real number ;
cluster min a1,a2 -> complex real ;
coherence
min r,s is real
by XXREAL_0:15;
cluster max a1,a2 -> complex real ;
coherence
max r,s is real
by XXREAL_0:16;
end;