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 for V being
Abelian add-associative right_zeroed right_complementable non empty ModuleStr
over R, F,G,H being FinSequence of V st len F = len G & len F =
len H & for k st k in dom F holds H.k = F/.k - G/.k holds Sum(H) = Sum(F) - Sum
  (G)
proof
  let R be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
  let V be Abelian add-associative right_zeroed right_complementable non
  empty ModuleStr over R, F,G,H be FinSequence of V;
  assume that
A1: len F = len G and
A2: len F = len H and
A3: for k st k in dom F holds H.k = F/.k - G/.k;
  deffunc F(Nat) = - G/.$1;
  consider I being FinSequence such that
A4: len I = len G and
A5: for k be Nat st k in dom I holds I.k = F(k) from FINSEQ_1:sch 2;
A6: dom I = Seg len G by A4,FINSEQ_1:def 3;
  then
A7: for k st k in Seg(len G) holds I.k = F(k) by A5;
  rng I c= the carrier of V
  proof
    let x be object;
    assume x in rng I;
    then consider y being object such that
A8: y in dom I and
A9: I.y = x by FUNCT_1:def 3;
    reconsider y as Element of NAT by A8;
    x = - G/.y by A5,A8,A9;
    then reconsider v = x as Element of V;
    v in V;
    hence thesis;
  end;
  then reconsider I as FinSequence of V by FINSEQ_1:def 4;
A10: Seg len G = dom G by FINSEQ_1:def 3;
  now
    let k;
A11: dom F = dom G by A1,FINSEQ_3:29;
    assume
A12: k in dom F;
    then k in dom I by A1,A4,FINSEQ_3:29;
    then
A13: I.k = I/.k by PARTFUN1:def 6;
    thus H.k = F/.k - G/.k by A3,A12
      .= F/.k + - G/.k
      .= F/.k + I/.k by A5,A6,A10,A12,A13,A11;
  end;
  then
A14: Sum(H) = Sum(F) + Sum(I) by A1,A2,A4,Th2;
  Sum(I) = - Sum(G) by A4,A7,A10,Th4;
  hence thesis by A14;
end;
