
theorem Th1:
  for A,B,A1,B1 be set st A misses B
  & A1 c= A & B1 c= B & A1 \/ B1 = A \/ B holds
  A1 = A & B1 = B
  proof
    let A,B,A1,B1 be set;
    assume A1: A misses B;
    assume A2: A1 c= A & B1 c= B;
    assume A3: A1 \/ B1 = A \/ B;
    A4:A c= A1
    proof
      let x be object;
      assume A5: x in A; then
      A6: x in A \/ B by XBOOLE_0:def 3;
      not x in B1
      proof
        assume x in B1; then
        x in A /\ B by A5,XBOOLE_0:def 4,A2;
        hence contradiction by A1, XBOOLE_0:def 7;
      end;
      hence x in A1 by A6,XBOOLE_0:def 3,A3;
    end;
    B c= B1
    proof
      let x be object;
      assume A7: x in B; then
      A8: x in A \/ B by XBOOLE_0:def 3;
      not x in A1
      proof
        assume x in A1; then
        x in A /\ B by A7,XBOOLE_0:def 4,A2;
        hence contradiction by A1, XBOOLE_0:def 7;
      end;
      hence x in B1 by A8,XBOOLE_0:def 3,A3;
    end;
    hence thesis by A2,XBOOLE_0:def 10,A4;
  end;
