reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;

theorem Th10:
  Sum(T \/ S) = Sum(T) + Sum(S) - Sum(T /\ S)
proof
  set A = S \ T;
  set B = T \ S;
  set Z = A \/ B;
  set I = T /\ S;
A1: A \/ I = S by XBOOLE_1:51;
A2: B \/ I = T by XBOOLE_1:51;
A3: Z = T \+\ S;
  then Z \/ I = T \/ S by XBOOLE_1:93;
  then Sum(T \/ S) + Sum(I) = Sum(Z) + Sum(I) + Sum(I) by A3,Th9,XBOOLE_1:103
    .= Sum(A) + Sum(B) + Sum(I) + Sum(I) by Th9,XBOOLE_1:82
    .= Sum(A) + (Sum(I) + Sum(B)) + Sum(I) by RLVECT_1:def 3
    .= (Sum(A) + Sum(I)) + (Sum(B) + Sum(I)) by RLVECT_1:def 3
    .= Sum(S) + (Sum(B) + Sum(I)) by A1,Th9,XBOOLE_1:89
    .= Sum(T) + Sum(S) by A2,Th9,XBOOLE_1:89;
  hence thesis by RLSUB_2:61;
end;
