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 Th43:
  {f}++{g} = {f+g}
proof
  let j;
  hereby
    assume j in {f}++{g};
    then consider w1,w2 such that
A1: j = w1+w2 and
A2: w1 in {f} & w2 in {g};
    w1 = f & w2 = g by A2,TARSKI:def 1;
    hence j in {f+g} by A1,TARSKI:def 1;
  end;
A3: f in {f} & g in {g} by TARSKI:def 1;
  assume j in {f+g};
  then
A4: j = f+g by TARSKI:def 1;
  f in ExtREAL & g in ExtREAL by XXREAL_0:def 1;
  hence thesis by A4,A3;
end;
