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 Th150:
  a ++ (A \ B) = (a++A) \ (a++B)
proof
  let z;
  hereby
    assume z in a ++ (A \ B);
    then consider c such that
A1: z = a+c and
A2: c in A\B by Th143;
A3: now
      assume a+c in a++B;
      then ex c1 st a+c = a+c1 & c1 in B by Th143;
      hence contradiction by A2,XBOOLE_0:def 5;
    end;
    a+c in a++A by A2,Th141;
    hence z in (a++A) \ (a++B) by A1,A3,XBOOLE_0:def 5;
  end;
  assume
A4: z in (a++A) \ (a++B);
  then consider c such that
A5: z = a+c and
A6: c in A by Th143;
  now
    assume not c in A\B;
    then c in B by A6,XBOOLE_0:def 5;
    then a+c in a++B by Th141;
    hence contradiction by A4,A5,XBOOLE_0:def 5;
  end;
  hence thesis by A5,Th141;
end;
