:: The Sum and Product of Finite Sequences of Real Numbers
:: by Czes{\l}aw Byli\'nski
::
:: Received May 11, 1990
:: Copyright (c) 1990 Association of Mizar Users



Lm1: for a, b being real number holds multreal . a,b = a * b
proof end;

Lm2: for a, b being complex number holds multcomplex . a,b = a * b
proof end;

registration
cluster real-valued set ;
existence
ex b1 being FinSequence st b1 is real-valued
proof end;
end;

definition
let F be real-valued Relation;
:: original: rng
redefine func rng F -> Subset of REAL ;
coherence
rng F is Subset of REAL by VALUED_0:def 3;
end;

registration
let D be non empty set ;
let F be Function of REAL ,D;
let F1 be real-valued FinSequence;
cluster F1 * F -> FinSequence-like ;
coherence
F * F1 is FinSequence-like
proof end;
end;

registration
let r be real number ;
cluster <*r*> -> real-valued ;
coherence
<*r*> is real-valued
proof end;
end;

registration
let r1, r2 be real number ;
cluster <*r1,r2*> -> real-valued ;
coherence
<*r1,r2*> is real-valued
proof end;
end;

registration
let r1, r2, r3 be real number ;
cluster <*r1,r2,r3*> -> real-valued ;
coherence
<*r1,r2,r3*> is real-valued
proof end;
end;

registration
let j be Nat;
let r be real number ;
cluster j |-> r -> real-valued ;
coherence
j |-> r is real-valued ;
end;

registration
let f, g be real-valued FinSequence;
cluster f ^ g -> real-valued ;
coherence
f ^ g is real-valued
proof end;
end;

theorem :: RVSUM_1:1
canceled;

theorem :: RVSUM_1:2
canceled;

theorem :: RVSUM_1:3
0 is_a_unity_wrt addreal by BINOP_2:2, SETWISEO:22;

definition
redefine func diffreal equals :: RVSUM_1:def 1
addreal * (id REAL ),compreal ;
compatibility
for b1 being Element of bool [:[:REAL ,REAL :],REAL :] holds
( b1 = diffreal iff b1 = addreal * (id REAL ),compreal )
proof end;
correctness
;
end;

:: deftheorem defines diffreal RVSUM_1:def 1 :
diffreal = addreal * (id REAL ),compreal ;

definition
func sqrreal -> UnOp of REAL means :Def2: :: RVSUM_1:def 2
for r being real number holds it . r = r ^2 ;
existence
ex b1 being UnOp of REAL st
for r being real number holds b1 . r = r ^2
proof end;
uniqueness
for b1, b2 being UnOp of REAL st ( for r being real number holds b1 . r = r ^2 ) & ( for r being real number holds b2 . r = r ^2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines sqrreal RVSUM_1:def 2 :
for b1 being UnOp of REAL holds
( b1 = sqrreal iff for r being real number holds b1 . r = r ^2 );

theorem :: RVSUM_1:4
canceled;

theorem :: RVSUM_1:5
canceled;

theorem :: RVSUM_1:6
canceled;

theorem :: RVSUM_1:7
canceled;

theorem :: RVSUM_1:8
canceled;

theorem :: RVSUM_1:9
canceled;

theorem :: RVSUM_1:10
canceled;

theorem :: RVSUM_1:11
canceled;

theorem :: RVSUM_1:12
canceled;

theorem :: RVSUM_1:13
1 is_a_unity_wrt multreal by BINOP_2:7, SETWISEO:22;

theorem :: RVSUM_1:14
canceled;

theorem :: RVSUM_1:15
canceled;

theorem Th16: :: RVSUM_1:16
multreal is_distributive_wrt addreal
proof end;

theorem Th17: :: RVSUM_1:17
sqrreal is_distributive_wrt multreal
proof end;

definition
let x be real number ;
func x multreal -> UnOp of REAL equals :: RVSUM_1:def 3
multreal [;] x,(id REAL );
coherence
multreal [;] x,(id REAL ) is UnOp of REAL
proof end;
end;

:: deftheorem defines multreal RVSUM_1:def 3 :
for x being real number holds x multreal = multreal [;] x,(id REAL );

Lm3: for r, r1 being real number holds (multreal [;] r,(id REAL )) . r1 = r * r1
proof end;

theorem :: RVSUM_1:18
canceled;

theorem :: RVSUM_1:19
for r, r1 being real number holds (r multreal ) . r1 = r * r1 by Lm3;

theorem :: RVSUM_1:20
for r being real number holds r multreal is_distributive_wrt addreal
proof end;

theorem Th21: :: RVSUM_1:21
compreal is_an_inverseOp_wrt addreal
proof end;

theorem Th22: :: RVSUM_1:22
addreal is having_an_inverseOp by Th21, FINSEQOP:def 2;

theorem Th23: :: RVSUM_1:23
the_inverseOp_wrt addreal = compreal by Th21, Th22, FINSEQOP:def 3;

theorem :: RVSUM_1:24
compreal is_distributive_wrt addreal by Th22, Th23, FINSEQOP:67;

Lm4: for F being real-valued FinSequence holds F is FinSequence of REAL
proof end;

definition
let F1, F2 be real-valued FinSequence;
:: original: +
redefine func F1 + F2 -> FinSequence of REAL equals :: RVSUM_1:def 4
addreal .: F1,F2;
coherence
F1 + F2 is FinSequence of REAL by Lm4;
compatibility
for b1 being FinSequence of REAL holds
( b1 = F1 + F2 iff b1 = addreal .: F1,F2 )
proof end;
commutativity
for b1 being FinSequence of REAL
for F1, F2 being real-valued FinSequence st b1 = addreal .: F1,F2 holds
b1 = addreal .: F2,F1
proof end;
end;

:: deftheorem defines + RVSUM_1:def 4 :
for F1, F2 being real-valued FinSequence holds F1 + F2 = addreal .: F1,F2;

definition
let i be Nat;
let R1, R2 be Element of i -tuples_on REAL ;
:: original: +
redefine func R1 + R2 -> Element of i -tuples_on REAL ;
coherence
R1 + R2 is Element of i -tuples_on REAL by FINSEQ_2:140;
end;

theorem :: RVSUM_1:25
canceled;

theorem :: RVSUM_1:26
canceled;

theorem Th27: :: RVSUM_1:27
for s being set
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds (R1 + R2) . s = (R1 . s) + (R2 . s)
proof end;

theorem :: RVSUM_1:28
for F being real-valued FinSequence holds (<*> REAL ) + F = <*> REAL
proof end;

theorem :: RVSUM_1:29
for r1, r2 being real number holds <*r1*> + <*r2*> = <*(r1 + r2)*>
proof end;

theorem :: RVSUM_1:30
for i being Nat
for r1, r2 being real number holds (i |-> r1) + (i |-> r2) = i |-> (r1 + r2)
proof end;

theorem :: RVSUM_1:31
canceled;

theorem Th32: :: RVSUM_1:32
for i being Nat
for R1, R2, R3 being Element of i -tuples_on REAL holds R1 + (R2 + R3) = (R1 + R2) + R3 by FINSEQOP:29;

theorem :: RVSUM_1:33
for i being Nat
for R being Element of i -tuples_on REAL holds R + (i |-> 0 ) = R by BINOP_2:2, FINSEQOP:57;

theorem :: RVSUM_1:34
canceled;

theorem Th35: :: RVSUM_1:35
for s being set
for F being real-valued FinSequence holds (- F) . s = - (F . s) by VALUED_1:8;

definition
let F be real-valued FinSequence;
:: original: -
redefine func - F -> FinSequence of REAL equals :: RVSUM_1:def 5
compreal * F;
coherence
- F is FinSequence of REAL by Lm4;
compatibility
for b1 being FinSequence of REAL holds
( b1 = - F iff b1 = compreal * F )
proof end;
end;

:: deftheorem defines - RVSUM_1:def 5 :
for F being real-valued FinSequence holds - F = compreal * F;

definition
let i be Nat;
let R be Element of i -tuples_on REAL ;
:: original: -
redefine func - R -> Element of i -tuples_on REAL ;
coherence
- R is Element of i -tuples_on REAL by FINSEQ_2:133;
end;

theorem :: RVSUM_1:36
for s being set
for i being Nat
for R being Element of i -tuples_on REAL holds (- R) . s = - (R . s) by Th35;

theorem :: RVSUM_1:37
- (<*> REAL ) = <*> REAL ;

theorem :: RVSUM_1:38
for r being real number holds - <*r*> = <*(- r)*>
proof end;

theorem Th39: :: RVSUM_1:39
for i being Nat
for r being real number holds - (i |-> r) = i |-> (- r)
proof end;

theorem :: RVSUM_1:40
for i being Nat
for R being Element of i -tuples_on REAL holds R + (- R) = i |-> 0 by Th22, Th23, BINOP_2:2, FINSEQOP:77;

theorem :: RVSUM_1:41
for i being Nat
for R1, R2 being Element of i -tuples_on REAL st R1 + R2 = i |-> 0 holds
R1 = - R2 by Th22, Th23, BINOP_2:2, FINSEQOP:78;

theorem :: RVSUM_1:42
canceled;

theorem :: RVSUM_1:43
for R1, R2 being complex-valued Function st - R1 = - R2 holds
R1 = R2
proof end;

theorem :: RVSUM_1:44
for i being Nat
for R1, R, R2 being Element of i -tuples_on REAL st R1 + R = R2 + R holds
R1 = R2
proof end;

theorem Th45: :: RVSUM_1:45
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds - (R1 + R2) = (- R1) + (- R2)
proof end;

definition
let F1, F2 be real-valued FinSequence;
:: original: -
redefine func F1 - F2 -> FinSequence of REAL equals :: RVSUM_1:def 6
diffreal .: F1,F2;
coherence
F1 - F2 is FinSequence of REAL by Lm4;
compatibility
for b1 being FinSequence of REAL holds
( b1 = F1 - F2 iff b1 = diffreal .: F1,F2 )
proof end;
end;

:: deftheorem defines - RVSUM_1:def 6 :
for F1, F2 being real-valued FinSequence holds F1 - F2 = diffreal .: F1,F2;

definition
let i be Nat;
let R1, R2 be Element of i -tuples_on REAL ;
:: original: -
redefine func R1 - R2 -> Element of i -tuples_on REAL ;
coherence
R1 - R2 is Element of i -tuples_on REAL by FINSEQ_2:140;
end;

theorem :: RVSUM_1:46
canceled;

theorem :: RVSUM_1:47
canceled;

theorem :: RVSUM_1:48
for s being set
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds (R1 - R2) . s = (R1 . s) - (R2 . s)
proof end;

theorem :: RVSUM_1:49
for F being real-valued FinSequence holds
( (<*> REAL ) - F = <*> REAL & F - (<*> REAL ) = <*> REAL )
proof end;

theorem :: RVSUM_1:50
for r1, r2 being real number holds <*r1*> - <*r2*> = <*(r1 - r2)*>
proof end;

theorem :: RVSUM_1:51
for i being Nat
for r1, r2 being real number holds (i |-> r1) - (i |-> r2) = i |-> (r1 - r2)
proof end;

theorem Th52: :: RVSUM_1:52
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds R1 - R2 = R1 + (- R2) by FINSEQOP:89;

theorem :: RVSUM_1:53
for i being Nat
for R being Element of i -tuples_on REAL holds R - (i |-> 0 ) = R
proof end;

theorem :: RVSUM_1:54
for i being Nat
for R being Element of i -tuples_on REAL holds (i |-> 0 ) - R = - R
proof end;

theorem :: RVSUM_1:55
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds R1 - (- R2) = R1 + R2
proof end;

theorem :: RVSUM_1:56
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds - (R1 - R2) = R2 - R1
proof end;

theorem Th57: :: RVSUM_1:57
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds - (R1 - R2) = (- R1) + R2
proof end;

theorem Th58: :: RVSUM_1:58
for i being Nat
for R being Element of i -tuples_on REAL holds R - R = i |-> 0
proof end;

theorem :: RVSUM_1:59
for i being Nat
for R1, R2 being Element of i -tuples_on REAL st R1 - R2 = i |-> 0 holds
R1 = R2
proof end;

theorem :: RVSUM_1:60
for i being Nat
for R1, R2, R3 being Element of i -tuples_on REAL holds (R1 - R2) - R3 = R1 - (R2 + R3)
proof end;

theorem Th61: :: RVSUM_1:61
for i being Nat
for R1, R2, R3 being Element of i -tuples_on REAL holds R1 + (R2 - R3) = (R1 + R2) - R3
proof end;

theorem :: RVSUM_1:62
for i being Nat
for R1, R2, R3 being Element of i -tuples_on REAL holds R1 - (R2 - R3) = (R1 - R2) + R3
proof end;

theorem :: RVSUM_1:63
for i being Nat
for R1, R being Element of i -tuples_on REAL holds R1 = (R1 + R) - R
proof end;

theorem :: RVSUM_1:64
for i being Nat
for R1, R being Element of i -tuples_on REAL holds R1 = (R1 - R) + R
proof end;

notation
let F be real-valued FinSequence;
let r be real number ;
synonym r * F for r (#) F;
end;

theorem :: RVSUM_1:65
canceled;

theorem Th66: :: RVSUM_1:66
for s being set
for r being real number
for F being real-valued FinSequence holds (r * F) . s = r * (F . s) by VALUED_1:6;

definition
let F be real-valued FinSequence;
let r be real number ;
:: original: *
redefine func r * F -> FinSequence of REAL equals :: RVSUM_1:def 7
(r multreal ) * F;
coherence
r * F is FinSequence of REAL by Lm4;
compatibility
for b1 being FinSequence of REAL holds
( b1 = r * F iff b1 = (r multreal ) * F )
proof end;
end;

:: deftheorem defines * RVSUM_1:def 7 :
for F being real-valued FinSequence
for r being real number holds r * F = (r multreal ) * F;

definition
let i be Nat;
let R be Element of i -tuples_on REAL ;
let r be real number ;
:: original: *
redefine func r * R -> Element of i -tuples_on REAL ;
coherence
r * R is Element of i -tuples_on REAL by FINSEQ_2:133;
end;

theorem :: RVSUM_1:67
for s being set
for i being Nat
for r being real number
for R being Element of i -tuples_on REAL holds (r * R) . s = r * (R . s) by Th66;

theorem :: RVSUM_1:68
for r being real number holds r * (<*> REAL ) = <*> REAL ;

theorem :: RVSUM_1:69
for r, r1 being real number holds r * <*r1*> = <*(r * r1)*>
proof end;

theorem Th70: :: RVSUM_1:70
for i being Nat
for r1, r2 being real number holds r1 * (i |-> r2) = i |-> (r1 * r2)
proof end;

theorem Th71: :: RVSUM_1:71
for i being Nat
for r1, r2 being real number
for R being Element of i -tuples_on REAL holds (r1 * r2) * R = r1 * (r2 * R)
proof end;

theorem :: RVSUM_1:72
for i being Nat
for r1, r2 being real number
for R being Element of i -tuples_on REAL holds (r1 + r2) * R = (r1 * R) + (r2 * R)
proof end;

theorem :: RVSUM_1:73
for i being Nat
for r being real number
for R1, R2 being Element of i -tuples_on REAL holds r * (R1 + R2) = (r * R1) + (r * R2)
proof end;

theorem :: RVSUM_1:74
for i being Nat
for R being Element of i -tuples_on REAL holds 1 * R = R
proof end;

theorem :: RVSUM_1:75
for i being Nat
for R being Element of i -tuples_on REAL holds 0 * R = i |-> 0
proof end;

theorem Th76: :: RVSUM_1:76
for i being Nat
for R being Element of i -tuples_on REAL holds (- 1) * R = - R
proof end;

notation
let F be real-valued FinSequence;
synonym sqr F for F ^2 ;
end;

definition
let F be real-valued FinSequence;
:: original: sqr
redefine func sqr F -> FinSequence of REAL equals :: RVSUM_1:def 8
sqrreal * F;
compatibility
for b1 being FinSequence of REAL holds
( b1 = sqr F iff b1 = sqrreal * F )
proof end;
coherence
sqr F is FinSequence of REAL by Lm4;
end;

:: deftheorem defines sqr RVSUM_1:def 8 :
for F being real-valued FinSequence holds sqr F = sqrreal * F;

definition
let i be Nat;
let R be Element of i -tuples_on REAL ;
:: original: sqr
redefine func sqr R -> Element of i -tuples_on REAL ;
coherence
sqr R is Element of i -tuples_on REAL by FINSEQ_2:133;
end;

theorem :: RVSUM_1:77
canceled;

theorem :: RVSUM_1:78
canceled;

theorem :: RVSUM_1:79
canceled;

theorem :: RVSUM_1:80
canceled;

theorem :: RVSUM_1:81
for r being real number holds sqr <*r*> = <*(r ^2 )*>
proof end;

theorem :: RVSUM_1:82
for i being Nat
for r being real number holds sqr (i |-> r) = i |-> (r ^2 )
proof end;

theorem Th83: :: RVSUM_1:83
for i being Nat
for R being Element of i -tuples_on REAL holds sqr (- R) = sqr R
proof end;

theorem Th84: :: RVSUM_1:84
for i being Nat
for r being real number
for R being Element of i -tuples_on REAL holds sqr (r * R) = (r ^2 ) * (sqr R)
proof end;

notation
let F1, F2 be real-valued FinSequence;
synonym mlt F1,F2 for F1 (#) F2;
end;

definition
let F1, F2 be real-valued FinSequence;
:: original: mlt
redefine func mlt F1,F2 -> FinSequence of REAL equals :: RVSUM_1:def 9
multreal .: F1,F2;
coherence
mlt F1,F2 is FinSequence of REAL by Lm4;
compatibility
for b1 being FinSequence of REAL holds
( b1 = mlt F1,F2 iff b1 = multreal .: F1,F2 )
proof end;
commutativity
for b1 being FinSequence of REAL
for F1, F2 being real-valued FinSequence st b1 = multreal .: F1,F2 holds
b1 = multreal .: F2,F1
proof end;
end;

:: deftheorem defines mlt RVSUM_1:def 9 :
for F1, F2 being real-valued FinSequence holds mlt F1,F2 = multreal .: F1,F2;

theorem :: RVSUM_1:85
canceled;

theorem Th86: :: RVSUM_1:86
for s being set
for F1, F2 being real-valued FinSequence st s in dom (mlt F1,F2) holds
(mlt F1,F2) . s = (F1 . s) * (F2 . s)
proof end;

definition
let i be Nat;
let R1, R2 be Element of i -tuples_on REAL ;
:: original: mlt
redefine func mlt R1,R2 -> Element of i -tuples_on REAL ;
coherence
mlt R1,R2 is Element of i -tuples_on REAL by FINSEQ_2:140;
end;

theorem Th87: :: RVSUM_1:87
for s being set
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds (mlt R1,R2) . s = (R1 . s) * (R2 . s)
proof end;

theorem :: RVSUM_1:88
for F being real-valued FinSequence holds mlt (<*> REAL ),F = <*> REAL
proof end;

theorem :: RVSUM_1:89
for r1, r2 being real number holds mlt <*r1*>,<*r2*> = <*(r1 * r2)*>
proof end;

theorem :: RVSUM_1:90
canceled;

theorem :: RVSUM_1:91
canceled;

theorem Th92: :: RVSUM_1:92
for i being Nat
for r being real number
for R being Element of i -tuples_on REAL holds mlt (i |-> r),R = r * R
proof end;

theorem :: RVSUM_1:93
for i being Nat
for r1, r2 being real number holds mlt (i |-> r1),(i |-> r2) = i |-> (r1 * r2)
proof end;

theorem Th94: :: RVSUM_1:94
for i being Nat
for r being real number
for R1, R2 being Element of i -tuples_on REAL holds r * (mlt R1,R2) = mlt (r * R1),R2
proof end;

theorem :: RVSUM_1:95
canceled;

theorem :: RVSUM_1:96
for i being Nat
for r being real number
for R being Element of i -tuples_on REAL holds r * R = mlt (i |-> r),R by Th92;

theorem Th97: :: RVSUM_1:97
for i being Nat
for R being Element of i -tuples_on REAL holds sqr R = mlt R,R
proof end;

theorem Th98: :: RVSUM_1:98
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds sqr (R1 + R2) = ((sqr R1) + (2 * (mlt R1,R2))) + (sqr R2)
proof end;

theorem Th99: :: RVSUM_1:99
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds sqr (R1 - R2) = ((sqr R1) - (2 * (mlt R1,R2))) + (sqr R2)
proof end;

theorem :: RVSUM_1:100
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds sqr (mlt R1,R2) = mlt (sqr R1),(sqr R2) by Th17, FINSEQOP:52;

notation
let F be Relation;
synonym complex-yielding F for complex-valued ;
end;

registration
cluster -> complex-yielding FinSequence of COMPLEX ;
coherence
for b1 being FinSequence of COMPLEX holds b1 is complex-yielding ;
cluster complex-yielding real-valued set ;
existence
ex b1 being FinSequence st
( b1 is real-valued & b1 is complex-yielding )
proof end;
end;

definition
let F be complex-yielding FinSequence;
canceled;
func Sum F -> complex number means :Def11: :: RVSUM_1:def 11
 ex f being FinSequence of COMPLEX st
( f = F & it = addcomplex $$ f );
existence
ex b1 being complex number ex f being FinSequence of COMPLEX st
( f = F & b1 = addcomplex $$ f )
proof end;
uniqueness
for b1, b2 being complex number st ex f being FinSequence of COMPLEX st
( f = F & b1 = addcomplex $$ f ) & ex f being FinSequence of COMPLEX st
( f = F & b2 = addcomplex $$ f ) holds
b1 = b2 ;
end;

:: deftheorem RVSUM_1:def 10 :
canceled;

:: deftheorem Def11 defines Sum RVSUM_1:def 11 :
for F being complex-yielding FinSequence
for b2 being complex number holds
( b2 = Sum F iff ex f being FinSequence of COMPLEX st
( f = F & b2 = addcomplex $$ f ) );

registration
let F be real-valued FinSequence;
cluster Sum F -> complex real ;
coherence
Sum F is real
proof end;
end;

theorem Th101: :: RVSUM_1:101
for F being FinSequence of REAL holds Sum F = addreal $$ F
proof end;

definition
let F be FinSequence of COMPLEX ;
:: original: Sum
redefine func Sum F -> Element of COMPLEX equals :: RVSUM_1:def 12
addcomplex $$ F;
coherence
Sum F is Element of COMPLEX by XCMPLX_0:def 2;
compatibility
for b1 being Element of COMPLEX holds
( b1 = Sum F iff b1 = addcomplex $$ F ) by Def11;
end;

:: deftheorem defines Sum RVSUM_1:def 12 :
for F being FinSequence of COMPLEX holds Sum F = addcomplex $$ F;

definition
let F be FinSequence of REAL ;
:: original: Sum
redefine func Sum F -> Element of REAL equals :: RVSUM_1:def 13
addreal $$ F;
coherence
Sum F is Element of REAL by XREAL_0:def 1;
compatibility
for b1 being Element of REAL holds
( b1 = Sum F iff b1 = addreal $$ F ) by Th101;
end;

:: deftheorem defines Sum RVSUM_1:def 13 :
for F being FinSequence of REAL holds Sum F = addreal $$ F;

theorem Th102: :: RVSUM_1:102
Sum (<*> REAL ) = 0 by BINOP_2:2, FINSOP_1:11;

theorem :: RVSUM_1:103
for r being real number holds Sum <*r*> = r
proof end;

theorem Th104: :: RVSUM_1:104
for r being real number
for F being real-valued FinSequence holds Sum (F ^ <*r*>) = (Sum F) + r
proof end;

theorem Th105: :: RVSUM_1:105
for F1, F2 being real-valued FinSequence holds Sum (F1 ^ F2) = (Sum F1) + (Sum F2)
proof end;

theorem :: RVSUM_1:106
for r being real number
for F being real-valued FinSequence holds Sum (<*r*> ^ F) = r + (Sum F)
proof end;

theorem Th107: :: RVSUM_1:107
for r1, r2 being real number holds Sum <*r1,r2*> = r1 + r2
proof end;

theorem :: RVSUM_1:108
for r1, r2, r3 being real number holds Sum <*r1,r2,r3*> = (r1 + r2) + r3
proof end;

theorem :: RVSUM_1:109
for R being Element of 0 -tuples_on REAL holds Sum R = 0 by Th102, FINSEQ_2:113;

theorem Th110: :: RVSUM_1:110
for i being Nat
for r being real number holds Sum (i |-> r) = i * r
proof end;

theorem Th111: :: RVSUM_1:111
for i being Nat holds Sum (i |-> 0 ) = 0
proof end;

theorem Th112: :: RVSUM_1:112
for i being Nat
for R1, R2 being Element of i -tuples_on REAL st ( for j being Nat st j in Seg i holds
R1 . j <= R2 . j ) holds
Sum R1 <= Sum R2
proof end;

theorem Th113: :: RVSUM_1:113
for i being Nat
for R1, R2 being Element of i -tuples_on REAL st ( for j being Nat st j in Seg i holds
R1 . j <= R2 . j ) & ex j being Nat st
( j in Seg i & R1 . j < R2 . j ) holds
Sum R1 < Sum R2
proof end;

theorem Th114: :: RVSUM_1:114
for F being real-valued FinSequence st ( for i being Nat st i in dom F holds
0 <= F . i ) holds
0 <= Sum F
proof end;

theorem Th115: :: RVSUM_1:115
for F being real-valued FinSequence st ( for i being Nat st i in dom F holds
0 <= F . i ) & ex i being Nat st
( i in dom F & 0 < F . i ) holds
0 < Sum F
proof end;

theorem Th116: :: RVSUM_1:116
for F being real-valued FinSequence holds 0 <= Sum (sqr F)
proof end;

theorem Th117: :: RVSUM_1:117
for r being real number
for F being real-valued FinSequence holds Sum (r * F) = r * (Sum F)
proof end;

theorem :: RVSUM_1:118
for F being real-valued FinSequence holds Sum (- F) = - (Sum F)
proof end;

theorem Th119: :: RVSUM_1:119
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds Sum (R1 + R2) = (Sum R1) + (Sum R2)
proof end;

theorem Th120: :: RVSUM_1:120
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds Sum (R1 - R2) = (Sum R1) - (Sum R2)
proof end;

theorem :: RVSUM_1:121
for i being Nat
for R being Element of i -tuples_on REAL st Sum (sqr R) = 0 holds
R = i |-> 0
proof end;

theorem :: RVSUM_1:122
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds (Sum (mlt R1,R2)) ^2 <= (Sum (sqr R1)) * (Sum (sqr R2))
proof end;

definition
let F be complex-yielding FinSequence;
func Product F -> complex number means :Def14: :: RVSUM_1:def 14
 ex f being FinSequence of COMPLEX st
( f = F & it = multcomplex $$ f );
existence
ex b1 being complex number ex f being FinSequence of COMPLEX st
( f = F & b1 = multcomplex $$ f )
proof end;
uniqueness
for b1, b2 being complex number st ex f being FinSequence of COMPLEX st
( f = F & b1 = multcomplex $$ f ) & ex f being FinSequence of COMPLEX st
( f = F & b2 = multcomplex $$ f ) holds
b1 = b2 ;
end;

:: deftheorem Def14 defines Product RVSUM_1:def 14 :
for F being complex-yielding FinSequence
for b2 being complex number holds
( b2 = Product F iff ex f being FinSequence of COMPLEX st
( f = F & b2 = multcomplex $$ f ) );

registration
let F be real-valued FinSequence;
cluster Product F -> complex real ;
coherence
Product F is real
proof end;
end;

theorem Th123: :: RVSUM_1:123
for F being FinSequence of REAL holds Product F = multreal $$ F
proof end;

definition
let F be FinSequence of COMPLEX ;
:: original: Product
redefine func Product F -> Element of COMPLEX equals :: RVSUM_1:def 15
multcomplex $$ F;
coherence
Product F is Element of COMPLEX by XCMPLX_0:def 2;
compatibility
for b1 being Element of COMPLEX holds
( b1 = Product F iff b1 = multcomplex $$ F ) by Def14;
end;

:: deftheorem defines Product RVSUM_1:def 15 :
for F being FinSequence of COMPLEX holds Product F = multcomplex $$ F;

definition
let F be FinSequence of REAL ;
:: original: Product
redefine func Product F -> Element of REAL equals :: RVSUM_1:def 16
multreal $$ F;
coherence
Product F is Element of REAL by XREAL_0:def 1;
compatibility
for b1 being Element of REAL holds
( b1 = Product F iff b1 = multreal $$ F ) by Th123;
end;

:: deftheorem defines Product RVSUM_1:def 16 :
for F being FinSequence of REAL holds Product F = multreal $$ F;

Lm5: for F being empty FinSequence holds Product F = 1
proof end;

theorem Th124: :: RVSUM_1:124
Product (<*> REAL ) = 1 by Lm5;

registration
let r be complex number ;
cluster <*r*> -> complex-yielding ;
coherence
<*r*> is complex-yielding
proof end;
end;

registration
let r1, r2 be complex number ;
cluster <*r1,r2*> -> complex-yielding ;
coherence
<*r1,r2*> is complex-yielding
proof end;
end;

registration
let r1, r2, r3 be complex number ;
cluster <*r1,r2,r3*> -> complex-yielding ;
coherence
<*r1,r2,r3*> is complex-yielding
proof end;
end;

theorem Th125: :: RVSUM_1:125
for r being complex number holds Product <*r*> = r
proof end;

registration
let f, g be complex-yielding FinSequence;
cluster f ^ g -> complex-yielding ;
coherence
f ^ g is complex-yielding
proof end;
end;

theorem Th126: :: RVSUM_1:126
for F being complex-yielding FinSequence
for r being complex number holds Product (F ^ <*r*>) = (Product F) * r
proof end;

theorem Th127: :: RVSUM_1:127
for F1, F2 being complex-yielding FinSequence holds Product (F1 ^ F2) = (Product F1) * (Product F2)
proof end;

theorem :: RVSUM_1:128
for r being real number
for F being real-valued FinSequence holds Product (<*r*> ^ F) = r * (Product F)
proof end;

theorem Th129: :: RVSUM_1:129
for r1, r2 being complex number holds Product <*r1,r2*> = r1 * r2
proof end;

theorem :: RVSUM_1:130
for r1, r2, r3 being complex number holds Product <*r1,r2,r3*> = (r1 * r2) * r3
proof end;

theorem :: RVSUM_1:131
for R being Element of 0 -tuples_on REAL holds Product R = 1 by Th124, FINSEQ_2:113;

theorem :: RVSUM_1:132
for i being Nat holds Product (i |-> 1) = 1
proof end;

Lm6: for p being complex-yielding FinSequence st len p <> 0 holds
ex q being complex-yielding FinSequence ex d being complex number st p = q ^ <*d*>
proof end;

theorem :: RVSUM_1:133
for F being complex-yielding FinSequence holds
( ex k being Nat st
( k in dom F & F . k = 0 ) iff Product F = 0 )
proof end;

theorem :: RVSUM_1:134
for i, j being Nat
for r being real number holds Product ((i + j) |-> r) = (Product (i |-> r)) * (Product (j |-> r))
proof end;

theorem :: RVSUM_1:135
for i, j being Nat
for r being real number holds Product ((i * j) |-> r) = Product (j |-> (Product (i |-> r)))
proof end;

theorem :: RVSUM_1:136
for i being Nat
for r1, r2 being real number holds Product (i |-> (r1 * r2)) = (Product (i |-> r1)) * (Product (i |-> r2))
proof end;

theorem Th137: :: RVSUM_1:137
for i being Nat
for R1, R2 being Element of i -tuples_on REAL holds Product (mlt R1,R2) = (Product R1) * (Product R2)
proof end;

theorem :: RVSUM_1:138
for i being Nat
for r being real number
for R being Element of i -tuples_on REAL holds Product (r * R) = (Product (i |-> r)) * (Product R)
proof end;

theorem :: RVSUM_1:139
for i being Nat
for R being Element of i -tuples_on REAL holds Product (sqr R) = (Product R) ^2
proof end;


definition
let i be Nat;
let z be Element of COMPLEX ;
:: original: |->
redefine func i |-> z -> FinSequence of COMPLEX ;
coherence
i |-> z is FinSequence of COMPLEX
proof end;
end;

theorem :: RVSUM_1:140
for F being FinSequence of COMPLEX holds Product F = multcomplex $$ F ;

theorem :: RVSUM_1:141
for i, j being Nat
for z being Element of COMPLEX holds Product ((i + j) |-> z) = (Product (i |-> z)) * (Product (j |-> z))
proof end;

theorem :: RVSUM_1:142
for i, j being Nat
for z being Element of COMPLEX holds Product ((i * j) |-> z) = Product (j |-> (Product (i |-> z))) by SETWOP_2:38;

theorem :: RVSUM_1:143
for i being Nat
for z1, z2 being Element of COMPLEX holds Product (i |-> (z1 * z2)) = (Product (i |-> z1)) * (Product (i |-> z2))
proof end;