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 Th54:
  F--G = {w1-w2: w1 in F & w2 in G}
proof
  thus F--G c= {w1-w2: w1 in F & w2 in G}
  proof
    let e be object;
    assume e in F--G;
    then consider w1,w2 such that
A1: e = w1+w2 and
A2: w1 in F and
A3: w2 in --G;
    e = w1- -w2 & -w2 in G by A1,A3,Th2;
    hence thesis by A2;
  end;
  let e be object;
  assume e in {w1-w2: w1 in F & w2 in G};
  then consider w1,w2 such that
A4: e = w1-w2 & w1 in F and
A5: w2 in G;
  -w2 in --G by A5;
  hence thesis by A4;
end;
