:: by Library Committee

::

:: Received February 11, 2003

:: Copyright (c) 2003-2011 Association of Mizar Users

begin

:: 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

coherence

not +infty is real

end;
coherence

not -infty is real

proof end;

cluster +infty -> non real ;coherence

not +infty is real

proof end;

registration

cluster natural -> real set ;

coherence

for b_{1} being number st b_{1} is natural holds

b_{1} is real

coherence

for b_{1} being number st b_{1} is real holds

b_{1} is complex

end;
coherence

for b

b

proof end;

cluster real -> complex set ;coherence

for b

b

proof end;

registration

cluster real set ;

existence

ex b_{1} being number st b_{1} is real

coherence

for b_{1} being number st b_{1} is real holds

b_{1} is ext-real

end;
existence

ex b

proof end;

cluster real -> ext-real set ;coherence

for b

b

proof end;

Lm1: 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 - x -> real ;

coherence

- x is real

coherence

x " is real

cluster x + y -> real ;

coherence

x + y is real

coherence

x * y is real

end;
cluster - x -> real ;

coherence

- x is real

proof end;

cluster x " -> real ;coherence

x " is real

proof end;

let y be real number ;cluster x + y -> real ;

coherence

x + y is real

proof end;

cluster x * y -> real ;coherence

x * y is real

proof end;

registration

let x, y be real number ;

cluster x - y -> real ;

coherence

x - y is real ;

cluster x / y -> real ;

coherence

x / y is real ;

end;
cluster x - y -> real ;

coherence

x - y is real ;

cluster x / y -> real ;

coherence

x / y is real ;

begin

Lm2: for r, s being real number st ( ( r in REAL+ & s in REAL+ & ex x9, y9 being Element of REAL+ st

( r = x9 & s = y9 & x9 <=' y9 ) ) or ( r in [:{0},REAL+:] & s in [:{0},REAL+:] & ex x9, y9 being Element of REAL+ st

( r = [0,x9] & s = [0,y9] & y9 <=' x9 ) ) or ( s in REAL+ & r in [:{0},REAL+:] ) ) holds

r <= s

proof end;

Lm3: {} in {{}}

by TARSKI:def 1;

Lm4: for r, s being real number st r <= s & s <= r holds

r = s

proof end;

Lm5: for r, s, t being real number st r <= s holds

r + t <= s + t

proof end;

Lm6: 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;

Lm7: 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 Lm8: 0 <= 1

by Lm2;

1 + (- 1) = 0

;

then consider x1, x2, y1, y2 being Element of REAL such that

Lm9: 1 = [*x1,x2*] and

Lm10: ( - 1 = [*y1,y2*] & 0 = [*(+ (x1,y1)),(+ (x2,y2))*] ) by XCMPLX_0:def 4;

Lm11: x1 = 1

by Lm1, Lm9;

Lm12: ( y1 = - 1 & + (x1,y1) = 0 )

by Lm1, Lm10;

Lm13: now

assume
- 1 in REAL+
; :: thesis: contradiction

then ex x9, y9 being Element of REAL+ st

( x1 = x9 & y1 = y9 & z = x9 + y9 ) by Lm11, Lm12, ARYTM_0:def 2, ARYTM_2:21;

hence contradiction by Lm11, ARYTM_2:6; :: thesis: verum

end;
then ex x9, y9 being Element of REAL+ st

( x1 = x9 & y1 = y9 & z = x9 + y9 ) by Lm11, Lm12, ARYTM_0:def 2, ARYTM_2:21;

hence contradiction by Lm11, ARYTM_2:6; :: thesis: verum

Lm14: for r, s being real number st r >= 0 & s > 0 holds

r + s > 0

proof end;

Lm15: 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;

Lm16: for r, s, t being real number st r <= s & 0 <= t holds

r * t <= s * t

proof end;

Lm17: for r, s being real number st r > 0 & s > 0 holds

r * s > 0

proof end;

Lm18: for r, s being real number st r > 0 & s < 0 holds

r * s < 0

proof end;

Lm19: for s, t being real number st s <= t holds

- t <= - s

proof end;

Lm20: 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 b_{1} being real number st b_{1} is positive

existence

ex b_{1} being real number st b_{1} is negative

existence

ex b_{1} being real number st b_{1} is empty

end;
existence

ex b

proof end;

cluster complex ext-real negative real set ;existence

ex b

proof end;

cluster zero complex ext-real real set ;existence

ex b

proof end;

registration

let r, s be non negative real number ;

cluster r + s -> non negative ;

coherence

not r + s is negative

end;
cluster r + s -> non negative ;

coherence

not r + s is negative

proof end;

registration

let r, s be non positive real number ;

cluster r + s -> non positive ;

coherence

not r + s is positive

end;
cluster r + s -> non positive ;

coherence

not r + s is positive

proof end;

registration

let r be positive real number ;

let s be non negative real number ;

cluster r + s -> positive ;

coherence

r + s is positive

coherence

s + r is positive ;

end;
let s be non negative real number ;

cluster r + s -> positive ;

coherence

r + s is positive

proof end;

cluster s + r -> positive ;coherence

s + r is positive ;

registration

let r be negative real number ;

let s be non positive real number ;

cluster r + s -> negative ;

coherence

r + s is negative

coherence

s + r is negative ;

end;
let s be non positive real number ;

cluster r + s -> negative ;

coherence

r + s is negative

proof end;

cluster s + r -> negative ;coherence

s + r is negative ;

registration

let r be non positive real number ;

cluster - r -> non negative ;

coherence

not - r is negative

end;
cluster - r -> non negative ;

coherence

not - r is negative

proof end;

registration

let r be non negative real number ;

cluster - r -> non positive ;

coherence

not - r is positive

end;
cluster - r -> non positive ;

coherence

not - r is positive

proof end;

registration

let r be non negative real number ;

let s be non positive real number ;

cluster r - s -> non negative ;

coherence

not r - s is negative ;

cluster s - r -> non positive ;

coherence

not s - r is positive ;

end;
let s be non positive real number ;

cluster r - s -> non negative ;

coherence

not r - s is negative ;

cluster s - r -> non positive ;

coherence

not s - r is positive ;

registration

let r be positive real number ;

let s be non positive real number ;

cluster r - s -> positive ;

coherence

r - s is positive ;

cluster s - r -> negative ;

coherence

s - r is negative ;

end;
let s be non positive real number ;

cluster r - s -> positive ;

coherence

r - s is positive ;

cluster s - r -> negative ;

coherence

s - r is negative ;

registration

let r be negative real number ;

let s be non negative real number ;

cluster r - s -> negative ;

coherence

r - s is negative ;

cluster s - r -> positive ;

coherence

s - r is positive ;

end;
let s be non negative real number ;

cluster r - s -> negative ;

coherence

r - s is negative ;

cluster s - r -> positive ;

coherence

s - r is positive ;

registration

let r be non positive real number ;

let s be non negative real number ;

cluster r * s -> non positive ;

coherence

not r * s is positive

coherence

not s * r is positive ;

end;
let s be non negative real number ;

cluster r * s -> non positive ;

coherence

not r * s is positive

proof end;

cluster s * r -> non positive ;coherence

not s * r is positive ;

registration

let r, s be non positive real number ;

cluster r * s -> non negative ;

coherence

not r * s is negative

end;
cluster r * s -> non negative ;

coherence

not r * s is negative

proof end;

registration

let r, s be non negative real number ;

cluster r * s -> non negative ;

coherence

not r * s is negative

end;
cluster r * s -> non negative ;

coherence

not r * s is negative

proof end;

registration
end;

registration

let r be non positive real number ;

cluster r " -> non positive ;

coherence

not r " is positive

end;
cluster r " -> non positive ;

coherence

not r " is positive

proof end;

registration
end;

registration

let r be non negative real number ;

cluster r " -> non negative ;

coherence

not r " is negative

end;
cluster r " -> non negative ;

coherence

not r " is negative

proof end;

registration

let r be non negative real number ;

let s be non positive real number ;

cluster r / s -> non positive ;

coherence

not r / s is positive ;

cluster s / r -> non positive ;

coherence

not s / r is positive ;

end;
let s be non positive real number ;

cluster r / s -> non positive ;

coherence

not r / s is positive ;

cluster s / r -> non positive ;

coherence

not s / r is positive ;

registration

let r, s be non negative real number ;

cluster r / s -> non negative ;

coherence

not r / s is negative ;

end;
cluster r / s -> non negative ;

coherence

not r / s is negative ;

registration

let r, s be non positive real number ;

cluster r / s -> non negative ;

coherence

not r / s is negative ;

end;
cluster r / s -> non negative ;

coherence

not r / s is negative ;

begin

registration

let r, s be real number ;

cluster min (r,s) -> real ;

coherence

min (r,s) is real by XXREAL_0:15;

cluster max (r,s) -> real ;

coherence

max (r,s) is real by XXREAL_0:16;

end;
cluster min (r,s) -> real ;

coherence

min (r,s) is real by XXREAL_0:15;

cluster max (r,s) -> real ;

coherence

max (r,s) is real by XXREAL_0:16;

begin

definition

let r, s be real number ;

func r -' s -> set equals :Def2: :: XREAL_0:def 2

r - s if r - s >= 0

otherwise 0 ;

correctness

coherence

( ( r - s >= 0 implies r - s is set ) & ( not r - s >= 0 implies 0 is set ) );

consistency

for b_{1} being set holds verum;

;

end;
func r -' s -> set equals :Def2: :: XREAL_0:def 2

r - s if r - s >= 0

otherwise 0 ;

correctness

coherence

( ( r - s >= 0 implies r - s is set ) & ( not r - s >= 0 implies 0 is set ) );

consistency

for b

;

:: deftheorem Def2 defines -' XREAL_0:def 2 :

for r, s being real number holds

( ( r - s >= 0 implies r -' s = r - s ) & ( not r - s >= 0 implies r -' s = 0 ) );

registration
end;

registration

let r, s be real number ;

cluster r -' s -> non negative real real number ;

coherence

for b_{1} being real number st b_{1} = r -' s holds

not b_{1} is negative

end;
cluster r -' s -> non negative real real number ;

coherence

for b

not b

proof end;