:: XREAL_0 semantic presentation
:: deftheorem Def1 defines real XREAL_0:def 1 :
Lemma24:
for x being real number
for x1, x2 being Element of REAL st x = [*x1,x2*] holds
( x2 = 0 & x = x1 )
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
Lemma51:
{} in {{} }
by TARSKI:def 1;
Lemma52:
for r, s being real number st r <= s & s <= r holds
r = s
Lemma61:
for r, s, t being real number st r <= s holds
r + t <= s + t
Lemma143:
for r, s, t being real number st r <= s & s <= t holds
r <= t
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 , ;
Lemma155:
for r, s being real number st r >= 0 & s > 0 holds
r + s > 0
Lemma156:
for r, s being real number st r <= 0 & s < 0 holds
r + s < 0
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
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 )
Lemma161:
for r, s being real number st r > 0 & s > 0 holds
r * s > 0
Lemma162:
for r, s being real number st r > 0 & s < 0 holds
r * s < 0
Lemma163:
for s, t being real number st s <= t holds
- t <= - s
Lemma164:
for r, s being real number st r <= 0 & s >= 0 holds
r * s <= 0
Lemma165:
for r being real number st r " = 0 holds
r = 0