theorem Th13:
  (add_inverse G)" = add_inverse G
proof
  set f = add_inverse G;
A1: dom f = the carrier of G by FUNCT_2:def 1;
   now
    let x be object;
    assume x in dom f;
    then reconsider g = x as Element of G;
A3: f.(-g) = - -g by Def6
      .= g;
    thus f.x = -g by Def6
      .= f".x by A1,A3,FUNCT_1:32;
  end;
  hence thesis by A1;
end;
