reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem Th21:
  (A ^^ B) \ (A ^^ C) c= A ^^ (B \ C) & (B ^^ A) \ (C ^^ A) c= (B \ C) ^^ A
proof
  thus (A ^^ B) \ (A ^^ C) c= A ^^ (B \ C)
  proof
    let x be object;
    assume
A1: x in (A ^^ B) \ (A ^^ C);
    then x in A ^^ B by XBOOLE_0:def 5;
    then consider a, b such that
A2: a in A and
A3: b in B and
A4: x = a ^ b by Def1;
A5: now
      assume not b in C;
      then b in B \ C by A3,XBOOLE_0:def 5;
      hence thesis by A2,A4,Def1;
    end;
    not x in A ^^ C by A1,XBOOLE_0:def 5;
    hence thesis by A2,A4,A5,Def1;
  end;
  thus (B ^^ A) \ (C ^^ A) c= (B \ C) ^^ A
  proof
    let x be object;
    assume
A6: x in (B ^^ A) \ (C ^^ A);
    then x in B ^^ A by XBOOLE_0:def 5;
    then consider b, a such that
A7: b in B and
A8: a in A & x = b ^ a by Def1;
A9: now
      assume not b in C;
      then b in B \ C by A7,XBOOLE_0:def 5;
      hence thesis by A8,Def1;
    end;
    not x in C ^^ A by A6,XBOOLE_0:def 5;
    hence thesis by A8,A9,Def1;
  end;
end;
