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 Th51:
  {a}++{b} = {a+b}
proof
  let z;
  hereby
    assume z in {a}++{b};
    then consider c1,c2 such that
A1: z = c1+c2 and
A2: c1 in {a} & c2 in {b};
    c1 = a & c2 = b by A2,TARSKI:def 1;
    hence z in {a+b} by A1,TARSKI:def 1;
  end;
A3: a in {a} & b in {b} by TARSKI:def 1;
  assume z in {a+b};
  then
A4: z = a+b by TARSKI:def 1;
  thus thesis by A4,A3;
end;
