:: SUPINF_2 semantic presentation
:: deftheorem Def1 defines 0. SUPINF_2:def 1 :
definition
let c
1, c
2 be
R_eal;
func c
1 + c
2 -> R_eal means :
Def2:
:: SUPINF_2:def 2
ex b
1, b
2 being
Real st
( a
1 = b
1 & a
2 = b
2 & a
3 = b
1 + b
2 )
if ( a
1 in REAL & a
2 in REAL )
a
3 = +infty if ( ( a
1 = +infty & a
2 <> -infty ) or ( a
2 = +infty & a
1 <> -infty ) )
a
3 = -infty if ( ( a
1 = -infty & a
2 <> +infty ) or ( a
2 = -infty & a
1 <> +infty ) )
otherwise a
3 = 0. ;
existence
( not ( c1 in REAL & c2 in REAL & ( for b1 being R_eal
for b2, b3 being Real holds
not ( c1 = b2 & c2 = b3 & b1 = b2 + b3 ) ) ) & not ( ( ( c1 = +infty & c2 <> -infty ) or ( c2 = +infty & c1 <> -infty ) ) & ( for b1 being R_eal holds
not b1 = +infty ) ) & not ( ( ( c1 = -infty & c2 <> +infty ) or ( c2 = -infty & c1 <> +infty ) ) & ( for b1 being R_eal holds
not b1 = -infty ) ) & not ( not ( c1 in REAL & c2 in REAL ) & not ( c1 = +infty & c2 <> -infty ) & not ( c2 = +infty & c1 <> -infty ) & not ( c1 = -infty & c2 <> +infty ) & not ( c2 = -infty & c1 <> +infty ) & ( for b1 being R_eal holds
not b1 = 0. ) ) )
uniqueness
for b1, b2 being R_eal holds
( ( c1 in REAL & c2 in REAL & ex b3, b4 being Real st
( c1 = b3 & c2 = b4 & b1 = b3 + b4 ) & ex b3, b4 being Real st
( c1 = b3 & c2 = b4 & b2 = b3 + b4 ) implies b1 = b2 ) & ( ( ( c1 = +infty & c2 <> -infty ) or ( c2 = +infty & c1 <> -infty ) ) & b1 = +infty & b2 = +infty implies b1 = b2 ) & ( ( ( c1 = -infty & c2 <> +infty ) or ( c2 = -infty & c1 <> +infty ) ) & b1 = -infty & b2 = -infty implies b1 = b2 ) & not ( not ( c1 in REAL & c2 in REAL ) & not ( c1 = +infty & c2 <> -infty ) & not ( c2 = +infty & c1 <> -infty ) & not ( c1 = -infty & c2 <> +infty ) & not ( c2 = -infty & c1 <> +infty ) & b1 = 0. & b2 = 0. & not b1 = b2 ) )
;
consistency
for b1 being R_eal holds
( ( c1 in REAL & c2 in REAL & ( ( c1 = +infty & c2 <> -infty ) or ( c2 = +infty & c1 <> -infty ) ) implies ( ex b2, b3 being Real st
( c1 = b2 & c2 = b3 & b1 = b2 + b3 ) iff b1 = +infty ) ) & ( c1 in REAL & c2 in REAL & ( ( c1 = -infty & c2 <> +infty ) or ( c2 = -infty & c1 <> +infty ) ) implies ( ex b2, b3 being Real st
( c1 = b2 & c2 = b3 & b1 = b2 + b3 ) iff b1 = -infty ) ) & ( ( ( c1 = +infty & c2 <> -infty ) or ( c2 = +infty & c1 <> -infty ) ) & ( ( c1 = -infty & c2 <> +infty ) or ( c2 = -infty & c1 <> +infty ) ) implies ( b1 = +infty iff b1 = -infty ) ) )
by XREAL_0:def 1;
commutativity
for b1, b2, b3 being R_eal holds
( not ( b2 in REAL & b3 in REAL & ( for b4, b5 being Real holds
not ( b2 = b4 & b3 = b5 & b1 = b4 + b5 ) ) ) & ( ( ( b2 = +infty & b3 <> -infty ) or ( b3 = +infty & b2 <> -infty ) ) implies b1 = +infty ) & ( ( ( b2 = -infty & b3 <> +infty ) or ( b3 = -infty & b2 <> +infty ) ) implies b1 = -infty ) & not ( not ( b2 in REAL & b3 in REAL ) & not ( b2 = +infty & b3 <> -infty ) & not ( b3 = +infty & b2 <> -infty ) & not ( b2 = -infty & b3 <> +infty ) & not ( b3 = -infty & b2 <> +infty ) & not b1 = 0. ) implies ( not ( b3 in REAL & b2 in REAL & ( for b4, b5 being Real holds
not ( b3 = b4 & b2 = b5 & b1 = b4 + b5 ) ) ) & ( ( ( b3 = +infty & b2 <> -infty ) or ( b2 = +infty & b3 <> -infty ) ) implies b1 = +infty ) & ( ( ( b3 = -infty & b2 <> +infty ) or ( b2 = -infty & b3 <> +infty ) ) implies b1 = -infty ) & not ( not ( b3 in REAL & b2 in REAL ) & not ( b3 = +infty & b2 <> -infty ) & not ( b2 = +infty & b3 <> -infty ) & not ( b3 = -infty & b2 <> +infty ) & not ( b2 = -infty & b3 <> +infty ) & not b1 = 0. ) ) )
;
end;
:: deftheorem Def2 defines + SUPINF_2:def 2 :
theorem Th1: :: SUPINF_2:1
for b
1, b
2 being
R_ealfor b
3, b
4 being
Real holds
( b
1 = b
3 & b
2 = b
4 implies b
1 + b
2 = b
3 + b
4 )
theorem Th2: :: SUPINF_2:2
:: deftheorem Def3 defines - SUPINF_2:def 3 :
:: deftheorem Def4 defines - SUPINF_2:def 4 :
for b
1, b
2 being
R_eal holds b
1 - b
2 = b
1 + (- b2);
theorem Th3: :: SUPINF_2:3
for b
1 being
R_ealfor b
2 being
Real holds
( b
1 = b
2 implies
- b
1 = - b
2 )
theorem Th4: :: SUPINF_2:4
theorem Th5: :: SUPINF_2:5
for b
1, b
2 being
R_ealfor b
3, b
4 being
Real holds
( b
1 = b
3 & b
2 = b
4 implies b
1 - b
2 = b
3 - b
4 )
theorem Th6: :: SUPINF_2:6
theorem Th7: :: SUPINF_2:7
theorem Th8: :: SUPINF_2:8
theorem Th9: :: SUPINF_2:9
theorem Th10: :: SUPINF_2:10
theorem Th11: :: SUPINF_2:11
theorem Th12: :: SUPINF_2:12
theorem Th13: :: SUPINF_2:13
theorem Th14: :: SUPINF_2:14
theorem Th15: :: SUPINF_2:15
Lemma18:
for b1 being R_eal holds
( - b1 in REAL iff b1 in REAL )
Lemma19:
for b1, b2 being R_eal holds
( b1 <= b2 implies - b2 <= - b1 )
theorem Th16: :: SUPINF_2:16
for b
1, b
2 being
R_eal holds
( b
1 <= b
2 iff
- b
2 <= - b
1 )
theorem Th17: :: SUPINF_2:17
for b
1, b
2 being
R_eal holds
( not ( b
1 < b
2 & not
- b
2 < - b
1 ) & not (
- b
2 < - b
1 & not b
1 < b
2 ) )
by Th16;
theorem Th18: :: SUPINF_2:18
theorem Th19: :: SUPINF_2:19
theorem Th20: :: SUPINF_2:20
for b
1, b
2, b
3 being
R_eal holds
(
0. <= b
3 &
0. <= b
1 & b
2 = b
1 + b
3 implies b
1 <= b
2 )
Lemma23:
for b1, b2, b3, b4 being R_eal holds
( 0. <= b1 & 0. <= b3 & b1 <= b2 & b3 <= b4 implies b1 + b3 <= b2 + b4 )
theorem Th21: :: SUPINF_2:21
:: deftheorem Def5 defines + SUPINF_2:def 5 :
:: deftheorem Def6 defines - SUPINF_2:def 6 :
theorem Th22: :: SUPINF_2:22
theorem Th23: :: SUPINF_2:23
theorem Th24: :: SUPINF_2:24
theorem Th25: :: SUPINF_2:25
theorem Th26: :: SUPINF_2:26
theorem Th27: :: SUPINF_2:27
theorem Th28: :: SUPINF_2:28
theorem Th29: :: SUPINF_2:29
theorem Th30: :: SUPINF_2:30
theorem Th31: :: SUPINF_2:31
theorem Th32: :: SUPINF_2:32
theorem Th33: :: SUPINF_2:33
:: deftheorem Def7 defines sup SUPINF_2:def 7 :
:: deftheorem Def8 defines inf SUPINF_2:def 8 :
:: deftheorem Def9 defines + SUPINF_2:def 9 :
theorem Th34: :: SUPINF_2:34
theorem Th35: :: SUPINF_2:35
theorem Th36: :: SUPINF_2:36
:: deftheorem Def10 defines - SUPINF_2:def 10 :
theorem Th37: :: SUPINF_2:37
theorem Th38: :: SUPINF_2:38
:: deftheorem Def11 defines bounded_above SUPINF_2:def 11 :
:: deftheorem Def12 defines bounded_below SUPINF_2:def 12 :
:: deftheorem Def13 defines bounded SUPINF_2:def 13 :
theorem Th39: :: SUPINF_2:39
theorem Th40: :: SUPINF_2:40
theorem Th41: :: SUPINF_2:41
theorem Th42: :: SUPINF_2:42
theorem Th43: :: SUPINF_2:43
theorem Th44: :: SUPINF_2:44
theorem Th45: :: SUPINF_2:45
theorem Th46: :: SUPINF_2:46
theorem Th47: :: SUPINF_2:47
theorem Th48: :: SUPINF_2:48
theorem Th49: :: SUPINF_2:49
theorem Th50: :: SUPINF_2:50
theorem Th51: :: SUPINF_2:51
theorem Th52: :: SUPINF_2:52
:: deftheorem Def14 defines denumerable SUPINF_2:def 14 :
:: deftheorem Def15 defines nonnegative SUPINF_2:def 15 :
:: deftheorem Def16 defines Num SUPINF_2:def 16 :
theorem Th53: :: SUPINF_2:53
:: deftheorem Def17 defines Ser SUPINF_2:def 17 :
theorem Th54: :: SUPINF_2:54
theorem Th55: :: SUPINF_2:55
theorem Th56: :: SUPINF_2:56
:: deftheorem Def18 defines Set_of_Series SUPINF_2:def 18 :
Lemma61:
for b1 being Function of NAT , ExtREAL holds
rng b1 is non empty Subset of ExtREAL
:: deftheorem Def19 defines SUM SUPINF_2:def 19 :
:: deftheorem Def20 defines is_sumable SUPINF_2:def 20 :
theorem Th57: :: SUPINF_2:57
:: deftheorem Def21 defines Ser SUPINF_2:def 21 :
:: deftheorem Def22 defines nonnegative SUPINF_2:def 22 :
:: deftheorem Def23 defines SUM SUPINF_2:def 23 :
theorem Th58: :: SUPINF_2:58
theorem Th59: :: SUPINF_2:59
theorem Th60: :: SUPINF_2:60
theorem Th61: :: SUPINF_2:61
theorem Th62: :: SUPINF_2:62
theorem Th63: :: SUPINF_2:63
theorem Th64: :: SUPINF_2:64
:: deftheorem Def24 defines summable SUPINF_2:def 24 :
theorem Th65: :: SUPINF_2:65
theorem Th66: :: SUPINF_2:66
theorem Th67: :: SUPINF_2:67
theorem Th68: :: SUPINF_2:68
theorem Th69: :: SUPINF_2:69
theorem Th70: :: SUPINF_2:70
theorem Th71: :: SUPINF_2:71