theorem Th19:
  for f being Element of .:(G,X) holds dom f = X & rng f c= the carrier of G
proof
  let f be Element of .:(G,X);
  reconsider f as Element of Funcs(X, carr(G)) by Th17;
  f = f;
  hence thesis by FUNCT_2:def 1,RELAT_1:def 19;
end;
