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;

theorem Th4:
  for V being Abelian add-associative right_zeroed
  right_complementable non empty addLoopStr, F,G being FinSequence of the
carrier of V st len F = len G & (for k st k in dom F holds G.k = - F/.k) holds
  Sum(G) = - Sum(F)
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr, F,G be FinSequence of the carrier of V;
  assume that
A1: len F = len G and
A2: for k st k in dom F holds G.k = - F/.k;
  now
    let k be Nat;
    let v be Element of V;
    assume that
A3: k in dom G and
A4: v = F.k;
A5: dom G = Seg len G & dom F = Seg len F by FINSEQ_1:def 3;
    then v = F/.k by A1,A3,A4,PARTFUN1:def 6;
    hence G.k = - v by A1,A2,A3,A5;
  end;
  hence thesis by A1,RLVECT_1:40;
end;
