:: VFUNCT_2 semantic presentation
definition
let c
1 be non
empty set ;
let c
2 be
ComplexNormSpace;
let c
3, c
4 be
PartFunc of c
1,the
carrier of c
2;
deffunc H
1(
Element of c
1)
-> Element of the
carrier of c
2 =
(c3 /. a1) + (c4 /. a1);
deffunc H
2(
Element of c
1)
-> Element of the
carrier of c
2 =
(c3 /. a1) - (c4 /. a1);
defpred S
1[
set ] means a
1 in (dom c3) /\ (dom c4);
set c
5 =
(dom c3) /\ (dom c4);
func c
3 + c
4 -> PartFunc of a
1,the
carrier of a
2 means :
Def1:
:: VFUNCT_2:def 1
(
dom a
5 = (dom a3) /\ (dom a4) & ( for b
1 being
Element of a
1 holds
( b
1 in dom a
5 implies a
5 /. b
1 = (a3 /. b1) + (a4 /. b1) ) ) );
existence
ex b1 being PartFunc of c1,the carrier of c2 st
( dom b1 = (dom c3) /\ (dom c4) & ( for b2 being Element of c1 holds
( b2 in dom b1 implies b1 /. b2 = (c3 /. b2) + (c4 /. b2) ) ) )
uniqueness
for b1, b2 being PartFunc of c1,the carrier of c2 holds
( dom b1 = (dom c3) /\ (dom c4) & ( for b3 being Element of c1 holds
( b3 in dom b1 implies b1 /. b3 = (c3 /. b3) + (c4 /. b3) ) ) & dom b2 = (dom c3) /\ (dom c4) & ( for b3 being Element of c1 holds
( b3 in dom b2 implies b2 /. b3 = (c3 /. b3) + (c4 /. b3) ) ) implies b1 = b2 )
func c
3 - c
4 -> PartFunc of a
1,the
carrier of a
2 means :
Def2:
:: VFUNCT_2:def 2
(
dom a
5 = (dom a3) /\ (dom a4) & ( for b
1 being
Element of a
1 holds
( b
1 in dom a
5 implies a
5 /. b
1 = (a3 /. b1) - (a4 /. b1) ) ) );
existence
ex b1 being PartFunc of c1,the carrier of c2 st
( dom b1 = (dom c3) /\ (dom c4) & ( for b2 being Element of c1 holds
( b2 in dom b1 implies b1 /. b2 = (c3 /. b2) - (c4 /. b2) ) ) )
uniqueness
for b1, b2 being PartFunc of c1,the carrier of c2 holds
( dom b1 = (dom c3) /\ (dom c4) & ( for b3 being Element of c1 holds
( b3 in dom b1 implies b1 /. b3 = (c3 /. b3) - (c4 /. b3) ) ) & dom b2 = (dom c3) /\ (dom c4) & ( for b3 being Element of c1 holds
( b3 in dom b2 implies b2 /. b3 = (c3 /. b3) - (c4 /. b3) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines + VFUNCT_2:def 1 :
:: deftheorem Def2 defines - VFUNCT_2:def 2 :
:: deftheorem Def3 defines (#) VFUNCT_2:def 3 :
:: deftheorem Def4 defines (#) VFUNCT_2:def 4 :
:: deftheorem Def5 defines ||. VFUNCT_2:def 5 :
:: deftheorem Def6 defines - VFUNCT_2:def 6 :
theorem Th1: :: VFUNCT_2:1
theorem Th2: :: VFUNCT_2:2
theorem Th3: :: VFUNCT_2:3
theorem Th4: :: VFUNCT_2:4
theorem Th5: :: VFUNCT_2:5
theorem Th6: :: VFUNCT_2:6
theorem Th7: :: VFUNCT_2:7
theorem Th8: :: VFUNCT_2:8
theorem Th9: :: VFUNCT_2:9
theorem Th10: :: VFUNCT_2:10
theorem Th11: :: VFUNCT_2:11
theorem Th12: :: VFUNCT_2:12
theorem Th13: :: VFUNCT_2:13
theorem Th14: :: VFUNCT_2:14
theorem Th15: :: VFUNCT_2:15
theorem Th16: :: VFUNCT_2:16
theorem Th17: :: VFUNCT_2:17
theorem Th18: :: VFUNCT_2:18
theorem Th19: :: VFUNCT_2:19
theorem Th20: :: VFUNCT_2:20
theorem Th21: :: VFUNCT_2:21
theorem Th22: :: VFUNCT_2:22
theorem Th23: :: VFUNCT_2:23
theorem Th24: :: VFUNCT_2:24
theorem Th25: :: VFUNCT_2:25
theorem Th26: :: VFUNCT_2:26
theorem Th27: :: VFUNCT_2:27
theorem Th28: :: VFUNCT_2:28
theorem Th29: :: VFUNCT_2:29
theorem Th30: :: VFUNCT_2:30
theorem Th31: :: VFUNCT_2:31
theorem Th32: :: VFUNCT_2:32
theorem Th33: :: VFUNCT_2:33
theorem Th34: :: VFUNCT_2:34
theorem Th35: :: VFUNCT_2:35
theorem Th36: :: VFUNCT_2:36
theorem Th37: :: VFUNCT_2:37
theorem Th38: :: VFUNCT_2:38
theorem Th39: :: VFUNCT_2:39
theorem Th40: :: VFUNCT_2:40
:: deftheorem Def7 defines is_bounded_on VFUNCT_2:def 7 :
theorem Th41: :: VFUNCT_2:41
theorem Th42: :: VFUNCT_2:42
theorem Th43: :: VFUNCT_2:43
theorem Th44: :: VFUNCT_2:44
theorem Th45: :: VFUNCT_2:45
theorem Th46: :: VFUNCT_2:46
theorem Th47: :: VFUNCT_2:47
theorem Th48: :: VFUNCT_2:48
theorem Th49: :: VFUNCT_2:49
theorem Th50: :: VFUNCT_2:50
theorem Th51: :: VFUNCT_2:51
theorem Th52: :: VFUNCT_2:52
theorem Th53: :: VFUNCT_2:53
theorem Th54: :: VFUNCT_2:54
theorem Th55: :: VFUNCT_2:55
theorem Th56: :: VFUNCT_2:56
theorem Th57: :: VFUNCT_2:57