theorem Th41:
  x in A * g iff ex h st x = h * g & h in A
proof
  thus x in A * g implies ex h st x = h * g & h in A
  proof
    assume x in A * g;
    then consider a,b such that
A1: x = a * b & a in A and
A2: b in {g};
    b = g by A2,TARSKI:def 1;
    hence thesis by A1;
  end;
  given h such that
A3: x = h * g & h in A;
  g in {g} by TARSKI:def 1;
  hence thesis by A3;
end;
