:: COMSEQ_1 semantic presentation
theorem Th1: :: COMSEQ_1:1
theorem Th2: :: COMSEQ_1:2
:: deftheorem Def1 defines non-zero COMSEQ_1:def 1 :
theorem Th3: :: COMSEQ_1:3
theorem Th4: :: COMSEQ_1:4
theorem Th5: :: COMSEQ_1:5
canceled;
theorem Th6: :: COMSEQ_1:6
theorem Th7: :: COMSEQ_1:7
definition
let C be non
empty set ;
let f1 be
PartFunc of
C,
COMPLEX ,
f2 be
PartFunc of
C,
COMPLEX ;
deffunc H1(
set )
-> Element of
COMPLEX =
(f1 /. a1) + (f2 /. a1);
defpred S1[
set ]
means a1 in (dom f1) /\ (dom f2);
set X =
(dom f1) /\ (dom f2);
func c2 + c3 -> PartFunc of
a1,
COMPLEX means :
Def2:
:: COMSEQ_1:def 2
(
dom it = (dom f1) /\ (dom f2) & ( for
c being
Element of
C st
c in dom it holds
it . c = (f1 /. c) + (f2 /. c) ) );
existence
ex b1 being PartFunc of C, COMPLEX st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 . c = (f1 /. c) + (f2 /. c) ) )
uniqueness
for b1, b2 being PartFunc of C, COMPLEX st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 . c = (f1 /. c) + (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b2 holds
b2 . c = (f1 /. c) + (f2 /. c) ) holds
b1 = b2
commutativity
for b1, f1, f2 being PartFunc of C, COMPLEX st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 . c = (f1 /. c) + (f2 /. c) ) holds
( dom b1 = (dom f2) /\ (dom f1) & ( for c being Element of C st c in dom b1 holds
b1 . c = (f2 /. c) + (f1 /. c) ) )
;
deffunc H2(
set )
-> Element of
COMPLEX =
(f1 /. a1) * (f2 /. a1);
func c2 (#) c3 -> PartFunc of
a1,
COMPLEX means :
Def3:
:: COMSEQ_1:def 3
(
dom it = (dom f1) /\ (dom f2) & ( for
c being
Element of
C st
c in dom it holds
it . c = (f1 /. c) * (f2 /. c) ) );
existence
ex b1 being PartFunc of C, COMPLEX st
( dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 . c = (f1 /. c) * (f2 /. c) ) )
uniqueness
for b1, b2 being PartFunc of C, COMPLEX st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 . c = (f1 /. c) * (f2 /. c) ) & dom b2 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b2 holds
b2 . c = (f1 /. c) * (f2 /. c) ) holds
b1 = b2
commutativity
for b1, f1, f2 being PartFunc of C, COMPLEX st dom b1 = (dom f1) /\ (dom f2) & ( for c being Element of C st c in dom b1 holds
b1 . c = (f1 /. c) * (f2 /. c) ) holds
( dom b1 = (dom f2) /\ (dom f1) & ( for c being Element of C st c in dom b1 holds
b1 . c = (f2 /. c) * (f1 /. c) ) )
;
end;
:: deftheorem Def2 defines + COMSEQ_1:def 2 :
:: deftheorem Def3 defines (#) COMSEQ_1:def 3 :
:: deftheorem Def4 defines + COMSEQ_1:def 4 :
:: deftheorem Def5 defines (#) COMSEQ_1:def 5 :
:: deftheorem Def6 defines (#) COMSEQ_1:def 6 :
:: deftheorem Def7 defines (#) COMSEQ_1:def 7 :
:: deftheorem Def8 defines - COMSEQ_1:def 8 :
:: deftheorem Def9 defines - COMSEQ_1:def 9 :
:: deftheorem Def10 defines - COMSEQ_1:def 10 :
:: deftheorem Def11 defines " COMSEQ_1:def 11 :
:: deftheorem Def12 defines /" COMSEQ_1:def 12 :
:: deftheorem Def13 defines |. COMSEQ_1:def 13 :
:: deftheorem Def14 defines |. COMSEQ_1:def 14 :
theorem Th8: :: COMSEQ_1:8
canceled;
theorem Th9: :: COMSEQ_1:9
theorem Th10: :: COMSEQ_1:10
canceled;
theorem Th11: :: COMSEQ_1:11
theorem Th12: :: COMSEQ_1:12
theorem Th13: :: COMSEQ_1:13
theorem Th14: :: COMSEQ_1:14
theorem Th15: :: COMSEQ_1:15
theorem Th16: :: COMSEQ_1:16
theorem Th17: :: COMSEQ_1:17
theorem Th18: :: COMSEQ_1:18
theorem Th19: :: COMSEQ_1:19
theorem Th20: :: COMSEQ_1:20
theorem Th21: :: COMSEQ_1:21
theorem Th22: :: COMSEQ_1:22
theorem Th23: :: COMSEQ_1:23
theorem Th24: :: COMSEQ_1:24
theorem Th25: :: COMSEQ_1:25
theorem Th26: :: COMSEQ_1:26
theorem Th27: :: COMSEQ_1:27
theorem Th28: :: COMSEQ_1:28
theorem Th29: :: COMSEQ_1:29
theorem Th30: :: COMSEQ_1:30
theorem Th31: :: COMSEQ_1:31
theorem Th32: :: COMSEQ_1:32
theorem Th33: :: COMSEQ_1:33
theorem Th34: :: COMSEQ_1:34
theorem Th35: :: COMSEQ_1:35
theorem Th36: :: COMSEQ_1:36
theorem Th37: :: COMSEQ_1:37
theorem Th38: :: COMSEQ_1:38
theorem Th39: :: COMSEQ_1:39
theorem Th40: :: COMSEQ_1:40
theorem Th41: :: COMSEQ_1:41
theorem Th42: :: COMSEQ_1:42
theorem Th43: :: COMSEQ_1:43
theorem Th44: :: COMSEQ_1:44
theorem Th45: :: COMSEQ_1:45
theorem Th46: :: COMSEQ_1:46
for
seq1,
seq,
seq1' being
Complex_Sequence holds
(
(seq1 /" seq) + (seq1' /" seq) = (seq1 + seq1') /" seq &
(seq1 /" seq) - (seq1' /" seq) = (seq1 - seq1') /" seq )
by , ;
theorem Th47: :: COMSEQ_1:47
theorem Th48: :: COMSEQ_1:48
theorem Th49: :: COMSEQ_1:49
theorem Th50: :: COMSEQ_1:50
theorem Th51: :: COMSEQ_1:51
theorem Th52: :: COMSEQ_1:52
theorem Th53: :: COMSEQ_1:53