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 Th177:
  A--a = {c-a: c in A}
proof
A1: A--a = {c1-c2: c1 in A & c2 in {a}} by Th65;
  thus A--a c= {c-a: c in A}
  proof
    let e be object;
    assume e in A--a;
    then consider c1,c2 such that
A2: e = c1-c2 & c1 in A and
A3: c2 in {a} by A1;
    c2 = a by A3,TARSKI:def 1;
    hence thesis by A2;
  end;
  let e be object;
  assume e in {c-a: c in A};
  then ex c st e = c-a & c in A;
  hence thesis by Th176;
end;
