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 Th139:
  r ++ (F \ G) = (r++F) \ (r++G)
proof
A1: r ++ (F \ G) c= (r++F) \ (r++G)
  proof
    let i;
    assume i in r ++ (F \ G);
    then consider w such that
A2: i = r+w and
A3: w in F\G by Th134;
A4: now
      assume r+w in r++G;
      then consider w1 such that
A5:   r+w = r+w1 and
A6:   w1 in G by Th134;
      w = w1 by A5,XXREAL_3:11;
      hence contradiction by A3,A6,XBOOLE_0:def 5;
    end;
    r+w in r++F by A3,Th132;
    hence thesis by A2,A4,XBOOLE_0:def 5;
  end;
  (r++F) \ (r++G) c= r ++ (F \ G) by Th138;
  hence thesis by A1;
end;
