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 Th15:
  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(S)
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr, S,T be finite Subset of V;
  (T \ S) misses S by XBOOLE_1:79;
  then
A1: (T \ S) /\ S = {}V;
  (T \ S) \/ S = T \/ S by XBOOLE_1:39;
  then Sum(T \/ S) = Sum(T \ S) + Sum(S) - Sum((T \ S) /\ S) by Th13;
  then Sum(T \/ S) = Sum(T \ S) + Sum(S) - 0.V by A1,Th8
    .= Sum(T \ S) + Sum(S);
  hence thesis by RLSUB_2:61;
end;
