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 Th23:
  (F \/ G)"" = (F"") \/ (G"")
proof
  let i;
  hereby
    assume i in (F\/G)"";
    then consider w such that
A1: i = w" and
A2: w in F \/ G;
    w in F or w in G by A2,XBOOLE_0:def 3;
    then w" in F"" or w" in G"";
    hence i in F"" \/ G"" by A1,XBOOLE_0:def 3;
  end;
  assume
A3: i in F"" \/ G"";
  per cases by A3,XBOOLE_0:def 3;
  suppose
    i in F"";
    then consider w such that
A4: i = w" and
A5: w in F;
    w in F\/G by A5,XBOOLE_0:def 3;
    hence thesis by A4;
  end;
  suppose
    i in G"";
    then consider w such that
A6: i = w" and
A7: w in G;
    w in F\/G by A7,XBOOLE_0:def 3;
    hence thesis by A6;
  end;
end;
