reserve w, w1, w2 for Element of ExtREAL;
reserve c, c1, c2 for Complex;
reserve A, B, C, D for complex-membered set;
reserve F, G, H, I for ext-real-membered set;
reserve a, b, s, t, z for Complex;
reserve f, g, h, i, j for ExtReal;
reserve r for Real;
reserve e for set;

theorem Th33:
  (A /\ B)"" = (A"") /\ (B"")
proof
  let a;
  hereby
    assume a in (A/\B)"";
    then
A1: a" in A/\B by Th29;
    then a" in B by XBOOLE_0:def 4;
    then
A2: a in B"" by Th29;
    a" in A by A1,XBOOLE_0:def 4;
    then a in A"" by Th29;
    hence a in (A"") /\ (B"") by A2,XBOOLE_0:def 4;
  end;
  assume
A3: a in (A"") /\ (B"");
  then a in B"" by XBOOLE_0:def 4;
  then
A4: a" in B by Th29;
  a in A"" by A3,XBOOLE_0:def 4;
  then a" in A by Th29;
  then a" in A/\B by A4,XBOOLE_0:def 4;
  hence thesis by Th29;
end;
