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
  for V being Abelian add-associative right_zeroed right_complementable
  non empty addLoopStr, F,G being FinSequence of the carrier of V for f being
  Permutation of dom F st G = F * f holds Sum(F) = Sum(G)
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr, F,G be FinSequence of the carrier of V;
  let f be Permutation of dom F;
  assume G = F * f;
  then len F = len G & for i st i in dom G holds G.i = F.(f.i) by FINSEQ_2:44
,FUNCT_1:12;
  hence thesis by Th6;
end;
