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 Th169:
  F--f = {w-f: w in F}
proof
  thus F--f c= {w-f: w in F}
  proof
    let e be object;
    assume e in F--f;
    then consider w1,w2 such that
A1: e = w1+w2 & w1 in F and
A2: w2 in --{f};
    -w2 in {f} by A2,Th2;
    then w1- -w2 = w1+w2 & -w2 = f by TARSKI:def 1;
    hence thesis by A1;
  end;
  let e be object;
  assume e in {w-f: w in F};
  then ex w st e = w-f & w in F;
  hence thesis by Th168;
end;
