theorem Th27:
  x in g + A iff ex h st x = g + h & h in A
proof
  thus x in g + A implies ex h st x = g + h & h in A
  proof
    assume x in g + A;
    then consider g1,g2 such that
A1: x = g1 + g2 and
A2: g1 in {g} and
A3: g2 in A;
    g1 = g by A2,TARSKI:def 1;
    hence thesis by A1,A3;
  end;
  given h such that
A4: x = g + h & h in A;
  g in {g} by TARSKI:def 1;
  hence thesis by A4;
end;
