reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;
reserve e,e1,e2 for Element of LinComb(V);
reserve x,y for set,
  k,n for Nat;

theorem
  for R being add-associative right_zeroed right_complementable Abelian
    associative well-unital distributive non empty doubleLoopStr,
  a being Element of R
  for V being Abelian add-associative right_zeroed right_complementable
    vector-distributive scalar-distributive scalar-associative scalar-unital
 non empty ModuleStr over R, v,u being Element of V holds a * Sum
  <* v,u *> = a * v + a * u
proof
  let R be add-associative right_zeroed right_complementable Abelian
    associative well-unital distributive non empty doubleLoopStr,
  a be Element of R;
  let V be Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over R, v,u be Element of V;
  thus a * Sum<* v,u *> = a * (v + u) by RLVECT_1:45
    .= a * v + a * u by VECTSP_1:def 14;
end;
