theorem Th18:
  (+a)/" = +(-a)
proof
  set f = +a, g = +(-a);
A1: now
    reconsider h = f as Function;
    let y be object;
    assume y in the carrier of G;
    then reconsider y1 = y as Element of G;
    rng f = the carrier of G by FUNCT_2:def 3;
    then
A2: y1 in rng f;
    dom f = the carrier of G by FUNCT_2:def 1;
    then
A3: g.y1 in dom f & f/".y1 in dom f;
    f.(g.y) = (g.y1)+a by Def2
      .= y1+(-a)+a by Def2
      .= y1+(-a+a) by RLVECT_1:def 3
      .= y1+(0_G) by Def5
      .= y by Def4
      .= h.(h".y) by A2,FUNCT_1:35
      .= f.(f/".y) by TOPS_2:def 4;
    hence f/".y = g.y by A3,FUNCT_1:def 4;
  end;
  thus thesis by A1;
end;
