
theorem Th20:
  for G be Group,
  q be set,
  F be associative Group-like multMagma-Family of {q},
  f being Function of G,product F st F = q .--> G &
  for x being Element of G holds f . x = q .--> x holds
  f is bijective
  proof
    let G be Group,
    q be set,
    F be associative Group-like multMagma-Family of {q},
    f be Function of G,product F;
    assume A1:F = q .--> G;
    assume A2: for x being Element of G holds f . x = q .--> x;
    A3: q in {q} by TARSKI:def 1;
    A4: the carrier of product F = product (Carrier F) by GROUP_7:def 2;
    ex R being 1-sorted st
    R = F . q & (Carrier F) . q = the carrier of R by PRALG_1:def 15,A3;
    then
    A5: (Carrier F) . q = the carrier of G by A1,FUNCOP_1:7,A3;
    A6: dom (Carrier F) = {q} by PARTFUN1:def 2;
    for x1,x2 be object st x1 in the carrier of G
    & x2 in the carrier of G & f.x1 = f.x2 holds x1 = x2
    proof
      let z1,z2 be object;
      assume A7: z1 in the carrier of G
      & z2 in the carrier of G & f.z1 = f.z2;
      then
      reconsider x1=z1,x2=z2 as Element of G;
      A8: f . x2 = q .--> x2 by A2;
      thus z1 = (q .--> x1) .q by FUNCOP_1:7,A3
      .=(q .--> x2) .q by A8,A2,A7
      .=z2 by FUNCOP_1:7,A3;
    end; then
    A9: f is one-to-one by FUNCT_2:19;
    for y be object st y in the carrier of product F
    ex x be object st x in the carrier of G & y = f.x
    proof
      let y be object;
      assume y in the carrier of product F; then
      consider g be Function such that
      A10: y = g & dom g = dom (Carrier F)
      & for z be object st z in dom (Carrier F)
      holds g.z in (Carrier F).z by CARD_3:def 5,A4;
      reconsider x = g.q as Element of G by A5,A10,A3,A6;
      A11: for z being object st z in dom g holds
      g . z = x by TARSKI:def 1,A10;
      take x;
      thus x in the carrier of G;
      thus y = (dom g) --> x by A11,FUNCOP_1:11,A10
      .= q .--> x by A10,PARTFUN1:def 2
      .= f . x by A2;
    end;
    then rng f = the carrier of product F by FUNCT_2:10;
    then f is onto by FUNCT_2:def 3;
    hence thesis by A9;
  end;
