Article RVSUM_1, MML version 4.99.1005

:: RVSUM_1:exreg 1
registration
  cluster Relation-like Function-like real-valued finite FinSequence-like set;
end;

:: RVSUM_1:funcnot 1 => RVSUM_1:func 1
definition
  let a1 be Relation-like real-valued set;
  redefine func rng a1 -> Element of bool REAL;
end;

:: RVSUM_1:funcreg 1
registration
  let a1 be non empty set;
  let a2 be Function-like quasi_total Relation of REAL,a1;
  let a3 be Relation-like Function-like real-valued FinSequence-like set;
  cluster a3 * a2 -> Relation-like FinSequence-like;
end;

:: RVSUM_1:funcreg 2
registration
  let a1 be real set;
  cluster <*a1*> -> real-valued;
end;

:: RVSUM_1:funcreg 3
registration
  let a1, a2 be real set;
  cluster <*a1,a2*> -> real-valued;
end;

:: RVSUM_1:funcreg 4
registration
  let a1, a2, a3 be real set;
  cluster <*a1,a2,a3*> -> real-valued;
end;

:: RVSUM_1:funcreg 5
registration
  let a1 be natural set;
  let a2 be real set;
  cluster a1 |-> a2 -> Relation-like Function-like real-valued FinSequence-like;
end;

:: RVSUM_1:funcreg 6
registration
  let a1, a2 be Relation-like Function-like real-valued FinSequence-like set;
  cluster a1 ^ a2 -> Relation-like Function-like real-valued FinSequence-like;
end;

:: RVSUM_1:th 3
theorem
0 is_a_unity_wrt addreal;

:: RVSUM_1:funcnot 2 => BINOP_2:func 34
definition
  func diffreal -> Function-like quasi_total Relation of [:REAL,REAL:],REAL equals
    addreal *(id REAL,compreal);
end;

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

:: RVSUM_1:funcnot 3 => RVSUM_1:func 2
definition
  func sqrreal -> Function-like quasi_total Relation of REAL,REAL means
    for b1 being real set holds
       it . b1 = b1 ^2;
end;

:: RVSUM_1:def 2
theorem
for b1 being Function-like quasi_total Relation of REAL,REAL holds
      b1 = sqrreal
   iff
      for b2 being real set holds
         b1 . b2 = b2 ^2;

:: RVSUM_1:th 13
theorem
1 is_a_unity_wrt multreal;

:: RVSUM_1:th 16
theorem
multreal is_distributive_wrt addreal;

:: RVSUM_1:th 17
theorem
sqrreal is_distributive_wrt multreal;

:: RVSUM_1:funcnot 4 => RVSUM_1:func 3
definition
  let a1 be real set;
  func A1 multreal -> Function-like quasi_total Relation of REAL,REAL equals
    multreal [;](a1,id REAL);
end;

:: RVSUM_1:def 3
theorem
for b1 being real set holds
   b1 multreal = multreal [;](b1,id REAL);

:: RVSUM_1:th 19
theorem
for b1, b2 being real set holds
b1 multreal . b2 = b1 * b2;

:: RVSUM_1:th 20
theorem
for b1 being real set holds
   b1 multreal is_distributive_wrt addreal;

:: RVSUM_1:th 21
theorem
compreal is_an_inverseOp_wrt addreal;

:: RVSUM_1:th 22
theorem
addreal is having_an_inverseOp(REAL);

:: RVSUM_1:th 23
theorem
the_inverseOp_wrt addreal = compreal;

:: RVSUM_1:th 24
theorem
compreal is_distributive_wrt addreal;

:: RVSUM_1:funcnot 5 => RVSUM_1:func 4
definition
  let a1, a2 be Relation-like Function-like real-valued FinSequence-like set;
  redefine func A1 + A2 -> FinSequence of REAL equals
    addreal .:(a1,a2);
  commutativity;
::  for a1, a2 being Relation-like Function-like real-valued FinSequence-like set holds
::  a1 + a2 = a2 + a1;
end;

:: RVSUM_1:def 4
theorem
for b1, b2 being Relation-like Function-like real-valued FinSequence-like set holds
b1 + b2 = addreal .:(b1,b2);

:: RVSUM_1:funcnot 6 => RVSUM_1:func 5
definition
  let a1 be natural set;
  let a2, a3 be Element of a1 -tuples_on REAL;
  redefine func a2 + a3 -> Element of a1 -tuples_on REAL;
  commutativity;
::  for a1 being natural set
::  for a2, a3 being Element of a1 -tuples_on REAL holds
::  a2 + a3 = a3 + a2;
end;

:: RVSUM_1:th 27
theorem
for b1, b2 being natural set
for b3, b4 being Element of b1 -tuples_on REAL holds
(b3 + b4) . b2 = (b3 . b2) + (b4 . b2);

:: RVSUM_1:th 28
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set holds
   (<*> REAL) + b1 = <*> REAL;

:: RVSUM_1:th 29
theorem
for b1, b2 being real set holds
<*b1*> + <*b2*> = <*b1 + b2*>;

:: RVSUM_1:th 30
theorem
for b1 being natural set
for b2, b3 being real set holds
(b1 |-> b2) + (b1 |-> b3) = b1 |-> (b2 + b3);

:: RVSUM_1:th 32
theorem
for b1 being natural set
for b2, b3, b4 being Element of b1 -tuples_on REAL holds
b2 + (b3 + b4) = (b2 + b3) + b4;

:: RVSUM_1:th 33
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   b2 + (b1 |-> 0) = b2;

:: RVSUM_1:th 35
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set
for b2 being set holds
   (- b1) . b2 = - (b1 . b2);

:: RVSUM_1:funcnot 7 => RVSUM_1:func 6
definition
  let a1 be Relation-like Function-like real-valued FinSequence-like set;
  redefine func - A1 -> FinSequence of REAL equals
    a1 * compreal;
  involutiveness;
::  for a1 being Relation-like Function-like real-valued FinSequence-like set holds
::     - - a1 = a1;
end;

:: RVSUM_1:def 5
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set holds
   - b1 = b1 * compreal;

:: RVSUM_1:funcnot 8 => RVSUM_1:func 7
definition
  let a1 be natural set;
  let a2 be Element of a1 -tuples_on REAL;
  redefine func - a2 -> Element of a1 -tuples_on REAL;
  involutiveness;
::  for a1 being natural set
::  for a2 being Element of a1 -tuples_on REAL holds
::     - - a2 = a2;
end;

:: RVSUM_1:th 36
theorem
for b1, b2 being natural set
for b3 being Element of b1 -tuples_on REAL holds
   (- b3) . b2 = - (b3 . b2);

:: RVSUM_1:th 37
theorem
- <*> REAL = <*> REAL;

:: RVSUM_1:th 38
theorem
for b1 being real set holds
   - <*b1*> = <*- b1*>;

:: RVSUM_1:th 39
theorem
for b1 being natural set
for b2 being real set holds
   - (b1 |-> b2) = b1 |-> - b2;

:: RVSUM_1:th 40
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   b2 + - b2 = b1 |-> 0;

:: RVSUM_1:th 41
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL
      st b2 + b3 = b1 |-> 0
   holds b2 = - b3;

:: RVSUM_1:th 43
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL
      st - b2 = - b3
   holds b2 = b3;

:: RVSUM_1:th 44
theorem
for b1 being natural set
for b2, b3, b4 being Element of b1 -tuples_on REAL
      st b2 + b3 = b4 + b3
   holds b2 = b4;

:: RVSUM_1:th 45
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
- (b2 + b3) = (- b2) + - b3;

:: RVSUM_1:funcnot 9 => RVSUM_1:func 8
definition
  let a1, a2 be Relation-like Function-like real-valued FinSequence-like set;
  redefine func A1 - A2 -> FinSequence of REAL equals
    diffreal .:(a1,a2);
end;

:: RVSUM_1:def 6
theorem
for b1, b2 being Relation-like Function-like real-valued FinSequence-like set holds
b1 - b2 = diffreal .:(b1,b2);

:: RVSUM_1:funcnot 10 => RVSUM_1:func 9
definition
  let a1 be natural set;
  let a2, a3 be Element of a1 -tuples_on REAL;
  redefine func a2 - a3 -> Element of a1 -tuples_on REAL;
end;

:: RVSUM_1:th 48
theorem
for b1, b2 being natural set
for b3, b4 being Element of b1 -tuples_on REAL holds
(b3 - b4) . b2 = (b3 . b2) - (b4 . b2);

:: RVSUM_1:th 49
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set holds
   (<*> REAL) - b1 = <*> REAL & b1 - <*> REAL = <*> REAL;

:: RVSUM_1:th 50
theorem
for b1, b2 being real set holds
<*b1*> - <*b2*> = <*b1 - b2*>;

:: RVSUM_1:th 51
theorem
for b1 being natural set
for b2, b3 being real set holds
(b1 |-> b2) - (b1 |-> b3) = b1 |-> (b2 - b3);

:: RVSUM_1:th 52
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
b2 - b3 = b2 + - b3;

:: RVSUM_1:th 53
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   b2 - (b1 |-> 0) = b2;

:: RVSUM_1:th 54
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   (b1 |-> 0) - b2 = - b2;

:: RVSUM_1:th 55
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
b2 - - b3 = b2 + b3;

:: RVSUM_1:th 56
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
- (b2 - b3) = b3 - b2;

:: RVSUM_1:th 57
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
- (b2 - b3) = (- b2) + b3;

:: RVSUM_1:th 58
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   b2 - b2 = b1 |-> 0;

:: RVSUM_1:th 59
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL
      st b2 - b3 = b1 |-> 0
   holds b2 = b3;

:: RVSUM_1:th 60
theorem
for b1 being natural set
for b2, b3, b4 being Element of b1 -tuples_on REAL holds
(b2 - b3) - b4 = b2 - (b3 + b4);

:: RVSUM_1:th 61
theorem
for b1 being natural set
for b2, b3, b4 being Element of b1 -tuples_on REAL holds
b2 + (b3 - b4) = (b2 + b3) - b4;

:: RVSUM_1:th 62
theorem
for b1 being natural set
for b2, b3, b4 being Element of b1 -tuples_on REAL holds
b2 - (b3 - b4) = (b2 - b3) + b4;

:: RVSUM_1:th 63
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
b2 = (b2 + b3) - b3;

:: RVSUM_1:th 64
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
b2 = (b2 - b3) + b3;

:: RVSUM_1:funcnot 11 => VALUED_1:func 24
notation
  let a1 be Relation-like Function-like real-valued FinSequence-like set;
  let a2 be real set;
  synonym a2 * a1 for a2 (#) a1;
end;

:: RVSUM_1:th 66
theorem
for b1 being real set
for b2 being Relation-like Function-like real-valued FinSequence-like set
for b3 being set holds
   (b1 (#) b2) . b3 = b1 * (b2 . b3);

:: RVSUM_1:funcnot 12 => RVSUM_1:func 10
definition
  let a1 be Relation-like Function-like real-valued FinSequence-like set;
  let a2 be real set;
  redefine func A2 * A1 -> FinSequence of REAL equals
    a1 * (a2 multreal);
end;

:: RVSUM_1:def 7
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set
for b2 being real set holds
   b2 * b1 = b1 * (b2 multreal);

:: RVSUM_1:funcnot 13 => RVSUM_1:func 11
definition
  let a1 be natural set;
  let a2 be Element of a1 -tuples_on REAL;
  let a3 be real set;
  redefine func a3 * a2 -> Element of a1 -tuples_on REAL;
end;

:: RVSUM_1:th 67
theorem
for b1, b2 being natural set
for b3 being real set
for b4 being Element of b1 -tuples_on REAL holds
   (b3 * b4) . b2 = b3 * (b4 . b2);

:: RVSUM_1:th 68
theorem
for b1 being real set holds
   b1 * <*> REAL = <*> REAL;

:: RVSUM_1:th 69
theorem
for b1, b2 being real set holds
b1 * <*b2*> = <*b1 * b2*>;

:: RVSUM_1:th 70
theorem
for b1 being natural set
for b2, b3 being real set holds
b2 * (b1 |-> b3) = b1 |-> (b2 * b3);

:: RVSUM_1:th 71
theorem
for b1 being natural set
for b2, b3 being real set
for b4 being Element of b1 -tuples_on REAL holds
   (b2 * b3) * b4 = b2 * (b3 * b4);

:: RVSUM_1:th 72
theorem
for b1 being natural set
for b2, b3 being real set
for b4 being Element of b1 -tuples_on REAL holds
   (b2 + b3) * b4 = (b2 * b4) + (b3 * b4);

:: RVSUM_1:th 73
theorem
for b1 being natural set
for b2 being real set
for b3, b4 being Element of b1 -tuples_on REAL holds
b2 * (b3 + b4) = (b2 * b3) + (b2 * b4);

:: RVSUM_1:th 74
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   1 * b2 = b2;

:: RVSUM_1:th 75
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   0 * b2 = b1 |-> 0;

:: RVSUM_1:th 76
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   (- 1) * b2 = - b2;

:: RVSUM_1:funcnot 14 => RVSUM_1:func 12
definition
  let a1 be Relation-like Function-like real-valued FinSequence-like set;
  func sqr A1 -> FinSequence of REAL equals
    a1 * sqrreal;
end;

:: RVSUM_1:def 8
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set holds
   sqr b1 = b1 * sqrreal;

:: RVSUM_1:th 77
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set holds
   dom sqr b1 = dom b1;

:: RVSUM_1:th 78
theorem
for b1 being natural set
for b2 being Relation-like Function-like real-valued FinSequence-like set holds
   (sqr b2) . b1 = (b2 . b1) ^2;

:: RVSUM_1:funcnot 15 => RVSUM_1:func 13
definition
  let a1 be natural set;
  let a2 be Element of a1 -tuples_on REAL;
  redefine func sqr a2 -> Element of a1 -tuples_on REAL;
end;

:: RVSUM_1:th 79
theorem
for b1, b2 being natural set
for b3 being Element of b1 -tuples_on REAL holds
   (sqr b3) . b2 = (b3 . b2) ^2;

:: RVSUM_1:th 81
theorem
for b1 being real set holds
   sqr <*b1*> = <*b1 ^2*>;

:: RVSUM_1:th 82
theorem
for b1 being natural set
for b2 being real set holds
   sqr (b1 |-> b2) = b1 |-> (b2 ^2);

:: RVSUM_1:th 83
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   sqr - b2 = sqr b2;

:: RVSUM_1:th 84
theorem
for b1 being natural set
for b2 being real set
for b3 being Element of b1 -tuples_on REAL holds
   sqr (b2 * b3) = b2 ^2 * sqr b3;

:: RVSUM_1:funcnot 16 => VALUED_1:func 18
notation
  let a1, a2 be Relation-like Function-like real-valued FinSequence-like set;
  synonym mlt(a1,a2) for a1 (#) a2;
end;

:: RVSUM_1:funcnot 17 => RVSUM_1:func 14
definition
  let a1, a2 be Relation-like Function-like real-valued FinSequence-like set;
  redefine func mlt(A1,A2) -> FinSequence of REAL equals
    multreal .:(a1,a2);
  commutativity;
::  for a1, a2 being Relation-like Function-like real-valued FinSequence-like set holds
::  mlt(a1,a2) = mlt(a2,a1);
end;

:: RVSUM_1:def 9
theorem
for b1, b2 being Relation-like Function-like real-valued FinSequence-like set holds
mlt(b1,b2) = multreal .:(b1,b2);

:: RVSUM_1:th 86
theorem
for b1 being natural set
for b2, b3 being Relation-like Function-like real-valued FinSequence-like set
      st b1 in dom mlt(b2,b3)
   holds (mlt(b2,b3)) . b1 = (b2 . b1) * (b3 . b1);

:: RVSUM_1:funcnot 18 => RVSUM_1:func 15
definition
  let a1 be natural set;
  let a2, a3 be Element of a1 -tuples_on REAL;
  redefine func mlt(a2,a3) -> Element of a1 -tuples_on REAL;
  commutativity;
::  for a1 being natural set
::  for a2, a3 being Element of a1 -tuples_on REAL holds
::  mlt(a2,a3) = mlt(a3,a2);
end;

:: RVSUM_1:th 87
theorem
for b1, b2 being natural set
for b3, b4 being Element of b1 -tuples_on REAL holds
(mlt(b3,b4)) . b2 = (b3 . b2) * (b4 . b2);

:: RVSUM_1:th 88
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set holds
   mlt(<*> REAL,b1) = <*> REAL;

:: RVSUM_1:th 89
theorem
for b1, b2 being real set holds
mlt(<*b1*>,<*b2*>) = <*b1 * b2*>;

:: RVSUM_1:th 92
theorem
for b1 being natural set
for b2 being real set
for b3 being Element of b1 -tuples_on REAL holds
   mlt(b1 |-> b2,b3) = b2 * b3;

:: RVSUM_1:th 93
theorem
for b1 being natural set
for b2, b3 being real set holds
mlt(b1 |-> b2,b1 |-> b3) = b1 |-> (b2 * b3);

:: RVSUM_1:th 94
theorem
for b1 being natural set
for b2 being real set
for b3, b4 being Element of b1 -tuples_on REAL holds
b2 * mlt(b3,b4) = mlt(b2 * b3,b4);

:: RVSUM_1:th 96
theorem
for b1 being natural set
for b2 being real set
for b3 being Element of b1 -tuples_on REAL holds
   b2 * b3 = mlt(b1 |-> b2,b3);

:: RVSUM_1:th 97
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   sqr b2 = mlt(b2,b2);

:: RVSUM_1:th 98
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
sqr (b2 + b3) = ((sqr b2) + (2 * mlt(b2,b3))) + sqr b3;

:: RVSUM_1:th 99
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
sqr (b2 - b3) = ((sqr b2) - (2 * mlt(b2,b3))) + sqr b3;

:: RVSUM_1:th 100
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
sqr mlt(b2,b3) = mlt(sqr b2,sqr b3);

:: RVSUM_1:attrnot 1 => VALUED_0:attr 1
notation
  let a1 be Relation-like set;
  synonym complex-yielding for complex-valued;
end;

:: RVSUM_1:condreg 1
registration
  cluster -> complex-valued (FinSequence of COMPLEX);
end;

:: RVSUM_1:exreg 2
registration
  cluster Relation-like Function-like complex-valued real-valued finite FinSequence-like set;
end;

:: RVSUM_1:funcnot 19 => RVSUM_1:func 16
definition
  let a1 be Relation-like Function-like complex-valued FinSequence-like set;
  func Sum A1 -> complex set means
    ex b1 being FinSequence of COMPLEX st
       b1 = a1 & it = addcomplex "**" b1;
end;

:: RVSUM_1:def 11
theorem
for b1 being Relation-like Function-like complex-valued FinSequence-like set
for b2 being complex set holds
      b2 = Sum b1
   iff
      ex b3 being FinSequence of COMPLEX st
         b3 = b1 & b2 = addcomplex "**" b3;

:: RVSUM_1:funcreg 7
registration
  let a1 be Relation-like Function-like real-valued FinSequence-like set;
  cluster Sum a1 -> complex real;
end;

:: RVSUM_1:th 101
theorem
for b1 being FinSequence of REAL holds
   Sum b1 = addreal "**" b1;

:: RVSUM_1:funcnot 20 => RVSUM_1:func 17
definition
  let a1 be FinSequence of COMPLEX;
  redefine func Sum A1 -> Element of COMPLEX equals
    addcomplex "**" a1;
end;

:: RVSUM_1:def 12
theorem
for b1 being FinSequence of COMPLEX holds
   Sum b1 = addcomplex "**" b1;

:: RVSUM_1:funcnot 21 => RVSUM_1:func 18
definition
  let a1 be FinSequence of REAL;
  redefine func Sum A1 -> Element of REAL equals
    addreal "**" a1;
end;

:: RVSUM_1:def 13
theorem
for b1 being FinSequence of REAL holds
   Sum b1 = addreal "**" b1;

:: RVSUM_1:th 102
theorem
Sum <*> REAL = 0;

:: RVSUM_1:th 103
theorem
for b1 being real set holds
   Sum <*b1*> = b1;

:: RVSUM_1:th 104
theorem
for b1 being real set
for b2 being Relation-like Function-like real-valued FinSequence-like set holds
   Sum (b2 ^ <*b1*>) = (Sum b2) + b1;

:: RVSUM_1:th 105
theorem
for b1, b2 being Relation-like Function-like real-valued FinSequence-like set holds
Sum (b1 ^ b2) = (Sum b1) + Sum b2;

:: RVSUM_1:th 106
theorem
for b1 being real set
for b2 being Relation-like Function-like real-valued FinSequence-like set holds
   Sum (<*b1*> ^ b2) = b1 + Sum b2;

:: RVSUM_1:th 107
theorem
for b1, b2 being real set holds
Sum <*b1,b2*> = b1 + b2;

:: RVSUM_1:th 108
theorem
for b1, b2, b3 being real set holds
Sum <*b1,b2,b3*> = (b1 + b2) + b3;

:: RVSUM_1:th 109
theorem
for b1 being Element of 0 -tuples_on REAL holds
   Sum b1 = 0;

:: RVSUM_1:th 110
theorem
for b1 being natural set
for b2 being real set holds
   Sum (b1 |-> b2) = b1 * b2;

:: RVSUM_1:th 111
theorem
for b1 being natural set holds
   Sum (b1 |-> 0) = 0;

:: RVSUM_1:th 112
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL
      st for b4 being natural set
              st b4 in Seg b1
           holds b2 . b4 <= b3 . b4
   holds Sum b2 <= Sum b3;

:: RVSUM_1:th 113
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL
      st (for b4 being natural set
               st b4 in Seg b1
            holds b2 . b4 <= b3 . b4) &
         (ex b4 being natural set st
            b4 in Seg b1 & b2 . b4 < b3 . b4)
   holds Sum b2 < Sum b3;

:: RVSUM_1:th 114
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set
      st for b2 being natural set
              st b2 in dom b1
           holds 0 <= b1 . b2
   holds 0 <= Sum b1;

:: RVSUM_1:th 115
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set
      st (for b2 being natural set
               st b2 in dom b1
            holds 0 <= b1 . b2) &
         (ex b2 being natural set st
            b2 in dom b1 & 0 < b1 . b2)
   holds 0 < Sum b1;

:: RVSUM_1:th 116
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set holds
   0 <= Sum sqr b1;

:: RVSUM_1:th 117
theorem
for b1 being real set
for b2 being Relation-like Function-like real-valued FinSequence-like set holds
   Sum (b1 * b2) = b1 * Sum b2;

:: RVSUM_1:th 118
theorem
for b1 being Relation-like Function-like real-valued FinSequence-like set holds
   Sum - b1 = - Sum b1;

:: RVSUM_1:th 119
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
Sum (b2 + b3) = (Sum b2) + Sum b3;

:: RVSUM_1:th 120
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
Sum (b2 - b3) = (Sum b2) - Sum b3;

:: RVSUM_1:th 121
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL
      st Sum sqr b2 = 0
   holds b2 = b1 |-> 0;

:: RVSUM_1:th 122
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
(Sum mlt(b2,b3)) ^2 <= (Sum sqr b2) * Sum sqr b3;

:: RVSUM_1:funcnot 22 => RVSUM_1:func 19
definition
  let a1 be Relation-like Function-like complex-valued FinSequence-like set;
  func Product A1 -> complex set means
    ex b1 being FinSequence of COMPLEX st
       b1 = a1 & it = multcomplex "**" b1;
end;

:: RVSUM_1:def 14
theorem
for b1 being Relation-like Function-like complex-valued FinSequence-like set
for b2 being complex set holds
      b2 = Product b1
   iff
      ex b3 being FinSequence of COMPLEX st
         b3 = b1 & b2 = multcomplex "**" b3;

:: RVSUM_1:funcreg 8
registration
  let a1 be Relation-like Function-like real-valued FinSequence-like set;
  cluster Product a1 -> complex real;
end;

:: RVSUM_1:th 123
theorem
for b1 being FinSequence of REAL holds
   Product b1 = multreal "**" b1;

:: RVSUM_1:funcnot 23 => RVSUM_1:func 20
definition
  let a1 be FinSequence of COMPLEX;
  redefine func Product A1 -> Element of COMPLEX equals
    multcomplex "**" a1;
end;

:: RVSUM_1:def 15
theorem
for b1 being FinSequence of COMPLEX holds
   Product b1 = multcomplex "**" b1;

:: RVSUM_1:funcnot 24 => RVSUM_1:func 21
definition
  let a1 be FinSequence of REAL;
  redefine func Product A1 -> Element of REAL equals
    multreal "**" a1;
end;

:: RVSUM_1:def 16
theorem
for b1 being FinSequence of REAL holds
   Product b1 = multreal "**" b1;

:: RVSUM_1:th 124
theorem
Product <*> REAL = 1;

:: RVSUM_1:funcreg 9
registration
  let a1 be complex set;
  cluster <*a1*> -> complex-valued;
end;

:: RVSUM_1:funcreg 10
registration
  let a1, a2 be complex set;
  cluster <*a1,a2*> -> complex-valued;
end;

:: RVSUM_1:funcreg 11
registration
  let a1, a2, a3 be complex set;
  cluster <*a1,a2,a3*> -> complex-valued;
end;

:: RVSUM_1:th 125
theorem
for b1 being complex set holds
   Product <*b1*> = b1;

:: RVSUM_1:funcreg 12
registration
  let a1, a2 be Relation-like Function-like complex-valued FinSequence-like set;
  cluster a1 ^ a2 -> Relation-like Function-like complex-valued FinSequence-like;
end;

:: RVSUM_1:th 126
theorem
for b1 being Relation-like Function-like complex-valued FinSequence-like set
for b2 being complex set holds
   Product (b1 ^ <*b2*>) = (Product b1) * b2;

:: RVSUM_1:th 127
theorem
for b1, b2 being Relation-like Function-like complex-valued FinSequence-like set holds
Product (b1 ^ b2) = (Product b1) * Product b2;

:: RVSUM_1:th 128
theorem
for b1 being real set
for b2 being Relation-like Function-like real-valued FinSequence-like set holds
   Product (<*b1*> ^ b2) = b1 * Product b2;

:: RVSUM_1:th 129
theorem
for b1, b2 being complex set holds
Product <*b1,b2*> = b1 * b2;

:: RVSUM_1:th 130
theorem
for b1, b2, b3 being complex set holds
Product <*b1,b2,b3*> = (b1 * b2) * b3;

:: RVSUM_1:th 131
theorem
for b1 being Element of 0 -tuples_on REAL holds
   Product b1 = 1;

:: RVSUM_1:th 132
theorem
for b1 being natural set holds
   Product (b1 |-> 1) = 1;

:: RVSUM_1:th 133
theorem
for b1 being Relation-like Function-like complex-valued FinSequence-like set holds
      ex b2 being natural set st
         b2 in dom b1 & b1 . b2 = 0
   iff
      Product b1 = 0;

:: RVSUM_1:th 134
theorem
for b1, b2 being natural set
for b3 being real set holds
   Product ((b1 + b2) |-> b3) = (Product (b1 |-> b3)) * Product (b2 |-> b3);

:: RVSUM_1:th 135
theorem
for b1, b2 being natural set
for b3 being real set holds
   Product ((b1 * b2) |-> b3) = Product (b2 |-> Product (b1 |-> b3));

:: RVSUM_1:th 136
theorem
for b1 being natural set
for b2, b3 being real set holds
Product (b1 |-> (b2 * b3)) = (Product (b1 |-> b2)) * Product (b1 |-> b3);

:: RVSUM_1:th 137
theorem
for b1 being natural set
for b2, b3 being Element of b1 -tuples_on REAL holds
Product mlt(b2,b3) = (Product b2) * Product b3;

:: RVSUM_1:th 138
theorem
for b1 being natural set
for b2 being real set
for b3 being Element of b1 -tuples_on REAL holds
   Product (b2 * b3) = (Product (b1 |-> b2)) * Product b3;

:: RVSUM_1:th 139
theorem
for b1 being natural set
for b2 being Element of b1 -tuples_on REAL holds
   Product sqr b2 = (Product b2) ^2;

:: RVSUM_1:funcnot 25 => RVSUM_1:func 22
definition
  let a1 be natural set;
  let a2 be Element of COMPLEX;
  redefine func a1 |-> a2 -> FinSequence of COMPLEX;
end;

:: RVSUM_1:th 140
theorem
for b1 being FinSequence of COMPLEX holds
   Product b1 = multcomplex "**" b1;

:: RVSUM_1:th 141
theorem
for b1, b2 being natural set
for b3 being Element of COMPLEX holds
   Product ((b1 + b2) |-> b3) = (Product (b1 |-> b3)) * Product (b2 |-> b3);

:: RVSUM_1:th 142
theorem
for b1, b2 being natural set
for b3 being Element of COMPLEX holds
   Product ((b1 * b2) |-> b3) = Product (b2 |-> Product (b1 |-> b3));

:: RVSUM_1:th 143
theorem
for b1 being natural set
for b2, b3 being Element of COMPLEX holds
Product (b1 |-> (b2 * b3)) = (Product (b1 |-> b2)) * Product (b1 |-> b3);