theorem Th12:
  rng add_inverse G = the carrier of G
proof
  set f = add_inverse G;
  thus rng f c= the carrier of G;
  let x be object;
A1: dom f = the carrier of G by FUNCT_2:def 1;
  assume x in the carrier of G;
  then reconsider a = x as Element of G;
  f.(-a) = - -a by Def6
    .= a;
  hence thesis by A1,FUNCT_1:def 3;
end;
