Article RMOD_4, MML version 4.99.1005

:: RMOD_4:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4, b5 being FinSequence of the carrier of b2
      st len b4 = len b5 &
         (for b6 being natural set
         for b7 being Element of the carrier of b2
               st b6 in proj1 b4 & b7 = b5 . b6
            holds b4 . b6 = b7 * b3)
   holds Sum b4 = (Sum b5) * b3;

:: RMOD_4:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1 holds
   (Sum <*> the carrier of b2) * b3 = 0. b2;

:: RMOD_4:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4, b5 being Element of the carrier of b2 holds
(Sum <*b4,b5*>) * b3 = (b4 * b3) + (b5 * b3);

:: RMOD_4:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4, b5, b6 being Element of the carrier of b2 holds
(Sum <*b4,b5,b6*>) * b3 = ((b4 * b3) + (b5 * b3)) + (b6 * b3);

:: RMOD_4:funcnot 1 => RMOD_4:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be finite Element of bool the carrier of a2;
  func Sum A3 -> Element of the carrier of a2 means
    ex b1 being FinSequence of the carrier of a2 st
       proj2 b1 = a3 & b1 is one-to-one & it = Sum b1;
end;

:: RMOD_4:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being finite Element of bool the carrier of b2
for b4 being Element of the carrier of b2 holds
      b4 = Sum b3
   iff
      ex b5 being FinSequence of the carrier of b2 st
         proj2 b5 = b3 & b5 is one-to-one & b4 = Sum b5;

:: RMOD_4:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1 holds
   Sum {} b2 = 0. b2;

:: RMOD_4:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b2 holds
   Sum {b3} = b3;

:: RMOD_4:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of the carrier of b2
      st b3 <> b4
   holds Sum {b3,b4} = b3 + b4;

:: RMOD_4:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4, b5 being Element of the carrier of b2
      st b3 <> b4 & b4 <> b5 & b3 <> b5
   holds Sum {b3,b4,b5} = (b3 + b4) + b5;

:: RMOD_4:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being finite Element of bool the carrier of b2
      st b3 misses b4
   holds Sum (b3 \/ b4) = (Sum b3) + Sum b4;

:: RMOD_4:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being finite Element of bool the carrier of b2 holds
Sum (b3 \/ b4) = ((Sum b3) + Sum b4) - Sum (b3 /\ b4);

:: RMOD_4:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being finite Element of bool the carrier of b2 holds
Sum (b3 /\ b4) = ((Sum b3) + Sum b4) - Sum (b3 \/ b4);

:: RMOD_4:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being finite Element of bool the carrier of b2 holds
Sum (b3 \ b4) = (Sum (b3 \/ b4)) - Sum b4;

:: RMOD_4:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being finite Element of bool the carrier of b2 holds
Sum (b3 \ b4) = (Sum b3) - Sum (b3 /\ b4);

:: RMOD_4:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being finite Element of bool the carrier of b2 holds
Sum (b3 \+\ b4) = (Sum (b3 \/ b4)) - Sum (b3 /\ b4);

:: RMOD_4:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being finite Element of bool the carrier of b2 holds
Sum (b3 \+\ b4) = (Sum (b3 \ b4)) + Sum (b4 \ b3);

:: RMOD_4:modenot 1 => RMOD_4:mode 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  mode Linear_Combination of A2 -> Element of Funcs(the carrier of a2,the carrier of a1) means
    ex b1 being finite Element of bool the carrier of a2 st
       for b2 being Element of the carrier of a2
             st not b2 in b1
          holds it . b2 = 0. a1;
end;

:: RMOD_4:dfs 2
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Element of Funcs(the carrier of a2,the carrier of a1);
To prove
     a3 is Linear_Combination of a2
it is sufficient to prove
  thus ex b1 being finite Element of bool the carrier of a2 st
       for b2 being Element of the carrier of a2
             st not b2 in b1
          holds a3 . b2 = 0. a1;

:: RMOD_4:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of Funcs(the carrier of b2,the carrier of b1) holds
      b3 is Linear_Combination of b2
   iff
      ex b4 being finite Element of bool the carrier of b2 st
         for b5 being Element of the carrier of b2
               st not b5 in b4
            holds b3 . b5 = 0. b1;

:: RMOD_4:funcnot 2 => RMOD_4:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Linear_Combination of a2;
  func Carrier A3 -> finite Element of bool the carrier of a2 equals
    {b1 where b1 is Element of the carrier of a2: a3 . b1 <> 0. a1};
end;

:: RMOD_4:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2 holds
   Carrier b3 = {b4 where b4 is Element of the carrier of b2: b3 . b4 <> 0. b1};

:: RMOD_4:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being set
for b4 being Linear_Combination of b2 holds
      b3 in Carrier b4
   iff
      ex b5 being Element of the carrier of b2 st
         b3 = b5 & b4 . b5 <> 0. b1;

:: RMOD_4:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b2
for b4 being Linear_Combination of b2 holds
      b4 . b3 = 0. b1
   iff
      not b3 in Carrier b4;

:: RMOD_4:funcnot 3 => RMOD_4:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  func ZeroLC A2 -> Linear_Combination of a2 means
    Carrier it = {};
end;

:: RMOD_4:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2 holds
      b3 = ZeroLC b2
   iff
      Carrier b3 = {};

:: RMOD_4:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b2 holds
   (ZeroLC b2) . b3 = 0. b1;

:: RMOD_4:modenot 2 => RMOD_4:mode 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Element of bool the carrier of a2;
  mode Linear_Combination of A3 -> Linear_Combination of a2 means
    Carrier it c= a3;
end;

:: RMOD_4:dfs 5
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Element of bool the carrier of a2;
  let a4 be Linear_Combination of a2;
To prove
     a4 is Linear_Combination of a3
it is sufficient to prove
  thus Carrier a4 c= a3;

:: RMOD_4:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2
for b4 being Linear_Combination of b2 holds
      b4 is Linear_Combination of b3
   iff
      Carrier b4 c= b3;

:: RMOD_4:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of bool the carrier of b2
for b5 being Linear_Combination of b3
      st b3 c= b4
   holds b5 is Linear_Combination of b4;

:: RMOD_4:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2 holds
   ZeroLC b2 is Linear_Combination of b3;

:: RMOD_4:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of {} the carrier of b2 holds
   b3 = ZeroLC b2;

:: RMOD_4:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2 holds
   b3 is Linear_Combination of Carrier b3;

:: RMOD_4:funcnot 4 => RMOD_4:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be FinSequence of the carrier of a2;
  let a4 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
  func A4 (#) A3 -> FinSequence of the carrier of a2 means
    len it = len a3 &
     (for b1 being natural set
           st b1 in proj1 it
        holds it . b1 = (a3 /. b1) * (a4 . (a3 /. b1)));
end;

:: RMOD_4:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being FinSequence of the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being FinSequence of the carrier of b2 holds
      b5 = b4 (#) b3
   iff
      len b5 = len b3 &
       (for b6 being natural set
             st b6 in proj1 b5
          holds b5 . b6 = (b3 /. b6) * (b4 . (b3 /. b6)));

:: RMOD_4:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being natural set
for b4 being Element of the carrier of b2
for b5 being FinSequence of the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
      st b3 in proj1 b5 & b4 = b5 . b3
   holds (b6 (#) b5) . b3 = b4 * (b6 . b4);

:: RMOD_4:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 holds
   b3 (#) <*> the carrier of b2 = <*> the carrier of b2;

:: RMOD_4:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 holds
   b4 (#) <*b3*> = <*b3 * (b4 . b3)*>;

:: RMOD_4:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 holds
   b5 (#) <*b3,b4*> = <*b3 * (b5 . b3),b4 * (b5 . b4)*>;

:: RMOD_4:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4, b5 being Element of the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 holds
   b6 (#) <*b3,b4,b5*> = <*b3 * (b6 . b3),b4 * (b6 . b4),b5 * (b6 . b5)*>;

:: RMOD_4:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being FinSequence of the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 holds
   b5 (#) (b3 ^ b4) = (b5 (#) b3) ^ (b5 (#) b4);

:: RMOD_4:funcnot 5 => RMOD_4:func 5
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Linear_Combination of a2;
  func Sum A3 -> Element of the carrier of a2 means
    ex b1 being FinSequence of the carrier of a2 st
       b1 is one-to-one & proj2 b1 = Carrier a3 & it = Sum (a3 (#) b1);
end;

:: RMOD_4:def 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of the carrier of b2 holds
      b4 = Sum b3
   iff
      ex b5 being FinSequence of the carrier of b2 st
         b5 is one-to-one & proj2 b5 = Carrier b3 & b4 = Sum (b3 (#) b5);

:: RMOD_4:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2
      st 0. b1 <> 1_ b1
   holds    b3 <> {} & b3 is linearly-closed(b1, b2)
   iff
      for b4 being Linear_Combination of b3 holds
         Sum b4 in b3;

:: RMOD_4:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1 holds
   Sum ZeroLC b2 = 0. b2;

:: RMOD_4:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of {} the carrier of b2 holds
   Sum b3 = 0. b2;

:: RMOD_4:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b2
for b4 being Linear_Combination of {b3} holds
   Sum b4 = b3 * (b4 . b3);

:: RMOD_4:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of the carrier of b2
   st b3 <> b4
for b5 being Linear_Combination of {b3,b4} holds
   Sum b5 = (b3 * (b5 . b3)) + (b4 * (b5 . b4));

:: RMOD_4:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2
      st Carrier b3 = {}
   holds Sum b3 = 0. b2;

:: RMOD_4:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b2
for b4 being Linear_Combination of b2
      st Carrier b4 = {b3}
   holds Sum b4 = b3 * (b4 . b3);

:: RMOD_4:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Linear_Combination of b2
      st Carrier b5 = {b3,b4} & b3 <> b4
   holds Sum b5 = (b3 * (b5 . b3)) + (b4 * (b5 . b4));

:: RMOD_4:prednot 1 => RMOD_4:pred 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3, a4 be Linear_Combination of a2;
  redefine pred A3 = A4 means
    for b1 being Element of the carrier of a2 holds
       a3 . b1 = a4 . b1;
  symmetry;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
::  for a2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1
::  for a3, a4 being Linear_Combination of a2
::        st a3 = a4
::     holds a4 = a3;
  reflexivity;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
::  for a2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1
::  for a3 being Linear_Combination of a2 holds
::     a3 = a3;
end;

:: RMOD_4:dfs 8
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3, a4 be Linear_Combination of a2;
To prove
     a3 = a4
it is sufficient to prove
  thus for b1 being Element of the carrier of a2 holds
       a3 . b1 = a4 . b1;

:: RMOD_4:def 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Linear_Combination of b2 holds
   b3 = b4
iff
   for b5 being Element of the carrier of b2 holds
      b3 . b5 = b4 . b5;

:: RMOD_4:funcnot 6 => RMOD_4:func 6
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3, a4 be Linear_Combination of a2;
  func A3 + A4 -> Linear_Combination of a2 means
    for b1 being Element of the carrier of a2 holds
       it . b1 = (a3 . b1) + (a4 . b1);
end;

:: RMOD_4:def 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4, b5 being Linear_Combination of b2 holds
   b5 = b3 + b4
iff
   for b6 being Element of the carrier of b2 holds
      b5 . b6 = (b3 . b6) + (b4 . b6);

:: RMOD_4:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Linear_Combination of b2 holds
Carrier (b3 + b4) c= (Carrier b3) \/ Carrier b4;

:: RMOD_4:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2
for b4, b5 being Linear_Combination of b2
      st b4 is Linear_Combination of b3 & b5 is Linear_Combination of b3
   holds b4 + b5 is Linear_Combination of b3;

:: RMOD_4:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Linear_Combination of b2 holds
b3 + b4 = b4 + b3;

:: RMOD_4:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4, b5 being Linear_Combination of b2 holds
b3 + (b4 + b5) = (b3 + b4) + b5;

:: RMOD_4:th 55
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2 holds
   b3 + ZeroLC b2 = b3 & (ZeroLC b2) + b3 = b3;

:: RMOD_4:funcnot 7 => RMOD_4:func 7
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Element of the carrier of a1;
  let a4 be Linear_Combination of a2;
  func A4 * A3 -> Linear_Combination of a2 means
    for b1 being Element of the carrier of a2 holds
       it . b1 = (a4 . b1) * a3;
end;

:: RMOD_4:def 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4, b5 being Linear_Combination of b2 holds
   b5 = b4 * b3
iff
   for b6 being Element of the carrier of b2 holds
      b5 . b6 = (b4 . b6) * b3;

:: RMOD_4:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4 being Linear_Combination of b2 holds
   Carrier (b4 * b3) c= Carrier b4;

:: RMOD_4:th 59
theorem
for b1 being non empty non degenerated right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of the carrier of b1
      st b4 <> 0. b1
   holds Carrier (b3 * b4) = Carrier b3;

:: RMOD_4:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2 holds
   b3 * 0. b1 = ZeroLC b2;

:: RMOD_4:th 61
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b2
for b5 being Linear_Combination of b2
      st b5 is Linear_Combination of b4
   holds b5 * b3 is Linear_Combination of b4;

:: RMOD_4:th 62
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of the carrier of b1
for b5 being Linear_Combination of b2 holds
   b5 * (b3 + b4) = (b5 * b3) + (b5 * b4);

:: RMOD_4:th 63
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b1
for b4, b5 being Linear_Combination of b2 holds
(b4 + b5) * b3 = (b4 * b3) + (b5 * b3);

:: RMOD_4:th 64
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Element of the carrier of b1
for b5 being Linear_Combination of b2 holds
   (b5 * b3) * b4 = b5 * (b3 * b4);

:: RMOD_4:th 65
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2 holds
   b3 * 1_ b1 = b3;

:: RMOD_4:funcnot 8 => RMOD_4:func 8
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3 be Linear_Combination of a2;
  func - A3 -> Linear_Combination of a2 equals
    a3 * - 1_ a1;
  involutiveness;
::  for a1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
::  for a2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1
::  for a3 being Linear_Combination of a2 holds
::     - - a3 = a3;
end;

:: RMOD_4:def 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2 holds
   - b3 = b3 * - 1_ b1;

:: RMOD_4:th 67
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b2
for b4 being Linear_Combination of b2 holds
   (- b4) . b3 = - (b4 . b3);

:: RMOD_4:th 68
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Linear_Combination of b2
      st b3 + b4 = ZeroLC b2
   holds b4 = - b3;

:: RMOD_4:th 69
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2 holds
   Carrier - b3 = Carrier b3;

:: RMOD_4:th 70
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2
for b4 being Linear_Combination of b2
      st b4 is Linear_Combination of b3
   holds - b4 is Linear_Combination of b3;

:: RMOD_4:funcnot 9 => RMOD_4:func 9
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over a1;
  let a3, a4 be Linear_Combination of a2;
  func A3 - A4 -> Linear_Combination of a2 equals
    a3 + - a4;
end;

:: RMOD_4:def 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Linear_Combination of b2 holds
b3 - b4 = b3 + - b4;

:: RMOD_4:th 73
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of the carrier of b2
for b4, b5 being Linear_Combination of b2 holds
(b4 - b5) . b3 = (b4 . b3) - (b5 . b3);

:: RMOD_4:th 74
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Linear_Combination of b2 holds
Carrier (b3 - b4) c= (Carrier b3) \/ Carrier b4;

:: RMOD_4:th 75
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Element of bool the carrier of b2
for b4, b5 being Linear_Combination of b2
      st b4 is Linear_Combination of b3 & b5 is Linear_Combination of b3
   holds b4 - b5 is Linear_Combination of b3;

:: RMOD_4:th 76
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2 holds
   b3 - b3 = ZeroLC b2;

:: RMOD_4:th 77
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Linear_Combination of b2 holds
Sum (b3 + b4) = (Sum b3) + Sum b4;

:: RMOD_4:th 78
theorem
for b1 being non empty non degenerated right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of the carrier of b1 holds
   Sum (b3 * b4) = (Sum b3) * b4;

:: RMOD_4:th 79
theorem
for b1 being non empty non degenerated right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3 being Linear_Combination of b2 holds
   Sum - b3 = - Sum b3;

:: RMOD_4:th 80
theorem
for b1 being non empty non degenerated right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive domRing-like doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed RightMod-like RightModStr over b1
for b3, b4 being Linear_Combination of b2 holds
Sum (b3 - b4) = (Sum b3) - Sum b4;