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, F,G being FinSequence of V
st len F = len G & for k st k in dom F holds G.k = a * F/.k holds Sum(G) =
  a * Sum(F)
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, F,G be FinSequence of V;
  assume that
A1: len F = len G and
A2: for k st k in dom F holds G.k = a * F/.k;
  now
    let k;
    let v be Element of V;
    assume that
A3: k in dom G and
A4: v = F.k;
A5: k in dom F by A1,A3,FINSEQ_3:29;
    then v = F/.k by A4,PARTFUN1:def 6;
    hence G.k = a * v by A2,A5;
  end;
  hence thesis by A1,Th66;
end;
