theorem Th28:
  for p being QC-formula of A holds Ex(x,p) is closed iff
  still_not-bound_in p c= {x}
proof
  let p be QC-formula of A;
  thus Ex(x,p) is closed implies still_not-bound_in p c= {x}
  proof
    assume still_not-bound_in Ex(x,p) = {};
    then {} = (still_not-bound_in p) \ {x} by Th19;
    hence thesis by XBOOLE_1:37;
  end;
  assume still_not-bound_in p c= {x};
  then {} = (still_not-bound_in p) \ {x} by XBOOLE_1:37;
  hence still_not-bound_in Ex(x,p) = {} by Th19;
end;
