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 Th194:
  a**A = {a*c: c in A}
proof
  thus a**A c= {a*c: c in A}
  proof
    let e be object;
    assume e in a**A;
    then consider c1,c2 such that
A1: e = c1*c2 and
A2: c1 in {a} and
A3: c2 in A;
    c1 = a by A2,TARSKI:def 1;
    hence thesis by A1,A3;
  end;
  let e be object;
  assume e in {a*c: c in A};
  then ex c st e = a*c & c in A;
  hence thesis by Th193;
end;
