:: XREAL_0 semantic presentation
:: deftheorem Def1 defines real XREAL_0:def 1 :
Lemma2:
for b1 being real number
for b2, b3 being Element of REAL holds
( b1 = [*b2,b3*] implies ( b3 = 0 & b1 = b2 ) )
Lemma3:
1 = succ 0
;
Lemma4:
for b1, b2 being real number holds
( b1 <= b2 implies ( not ( b1 in REAL+ & b2 in REAL+ & ( for b3, b4 being Element of REAL+ holds
not ( b1 = b3 & b2 = b4 & b3 <=' b4 ) ) ) & not ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] & ( for b3, b4 being Element of REAL+ holds
not ( b1 = [0,b3] & b2 = [0,b4] & b4 <=' b3 ) ) ) & ( not ( b1 in REAL+ & b2 in REAL+ ) & not ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] ) implies ( b2 in REAL+ & b1 in [:{0},REAL+ :] ) ) ) )
by XXREAL_0:def 5;
Lemma5:
for b1, b2 being real number holds
( not ( not ( b1 in REAL+ & b2 in REAL+ & ex b3, b4 being Element of REAL+ st
( b1 = b3 & b2 = b4 & b3 <=' b4 ) ) & not ( b1 in [:{0},REAL+ :] & b2 in [:{0},REAL+ :] & ex b3, b4 being Element of REAL+ st
( b1 = [0,b3] & b2 = [0,b4] & b4 <=' b3 ) ) & not ( b2 in REAL+ & b1 in [:{0},REAL+ :] ) ) implies b1 <= b2 )
Lemma6:
{} in {{} }
by TARSKI:def 1;
Lemma7:
for b1, b2 being real number holds
( b1 <= b2 & b2 <= b1 implies b1 = b2 )
Lemma8:
for b1, b2, b3 being real number holds
( b1 <= b2 implies b1 + b3 <= b2 + b3 )
Lemma9:
for b1, b2, b3 being real number holds
( b1 <= b2 & b2 <= b3 implies b1 <= b3 )
reconsider c1 = 0 as Element of REAL+ by ARYTM_2:21;
Lemma10:
not 0 in [:{0},REAL+ :]
by ARYTM_0:5, ARYTM_2:21, XBOOLE_0:3;
reconsider c2 = 1 as Element of REAL+ by Lemma3, ARYTM_2:21;
c1 <=' c2
by ARYTM_1:6;
then Lemma11:
0 <= 1
by Lemma5;
1 + (- 1) = 0
;
then consider c3, c4, c5, c6 being Element of REAL such that
Lemma12:
1 = [*c3,c4*]
and
Lemma13:
- 1 = [*c5,c6*]
and
Lemma14:
0 = [*(+ c3,c5),(+ c4,c6)*]
by XCMPLX_0:def 4;
Lemma15:
c3 = 1
by Lemma2, Lemma12;
Lemma16:
c5 = - 1
by Lemma2, Lemma13;
Lemma17:
+ c3,c5 = 0
by Lemma2, Lemma14;
Lemma19:
for b1, b2 being real number holds
not ( b1 >= 0 & b2 > 0 & not b1 + b2 > 0 )
Lemma20:
for b1, b2 being real number holds
not ( b1 <= 0 & b2 < 0 & not b1 + b2 < 0 )
reconsider c7 = 0 as Element of REAL+ by ARYTM_2:21;
Lemma21:
for b1, b2, b3 being real number holds
( b1 <= b2 & 0 <= b3 implies b1 * b3 <= b2 * b3 )
Lemma22:
for b1, b2, b3 being real number holds (b1 * b2) * b3 = b1 * (b2 * b3)
;
Lemma23:
for b1, b2 being real number holds
not ( b1 * b2 = 0 & not b1 = 0 & not b2 = 0 )
Lemma24:
for b1, b2 being real number holds
not ( b1 > 0 & b2 > 0 & not b1 * b2 > 0 )
Lemma25:
for b1, b2 being real number holds
not ( b1 > 0 & b2 < 0 & not b1 * b2 < 0 )
Lemma26:
for b1, b2 being real number holds
( b1 <= b2 implies - b2 <= - b1 )
Lemma27:
for b1, b2 being real number holds
( b1 <= 0 & b2 >= 0 implies b1 * b2 <= 0 )
Lemma28:
for b1 being real number holds
( b1 " = 0 implies b1 = 0 )