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 be Abelian add-associative right_zeroed right_complementable
non empty addLoopStr, S,T be finite Subset of V holds Sum(T \+\ S) = Sum(T \/
  S) - Sum(T /\ S)
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr, S,T be finite Subset of V;
  T \+\ S = (T \/ S) \ (T /\ S) by XBOOLE_1:101;
  hence Sum(T \+\ S) = Sum(T \/ S) - Sum((T \/ S) /\ (T /\ S)) by Th16
    .= Sum(T \/ S) - Sum((T \/ S) /\ T /\ S) by XBOOLE_1:16
    .= Sum(T \/ S) - Sum(T /\ S) by XBOOLE_1:21;
end;
