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;
