
theorem Th21:
  for q be set,
  F be associative Group-like multMagma-Family of {q},
  G be Group st F = q .--> G holds
  ex I be Homomorphism of G,product F st
  I is bijective &
  for x being Element of G holds I . x = q .--> x
  proof
    let q be set,
    F be associative Group-like multMagma-Family of {q},
    G be Group;
    assume A1: F = q .--> G;
    A2: q in {q} by TARSKI:def 1;
    A3: 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,A2;
    then
    A4: (Carrier F) . q = the carrier of G by A1,FUNCOP_1:7,A2;
    A5: dom (Carrier F) = {q} by PARTFUN1:def 2;
    defpred P[set, set] means $2= (q .--> $1);
    A6:for z be Element of G ex w be Element of product F st P[z,w]
    proof
      let z be Element of G;
      set w = q .--> z;
      now let i be object;
        assume A8:i in dom w; then
        A9: i = q by TARSKI:def 1;
        w.i = z by FUNCOP_1:7,A8;
        hence w.i in (Carrier F) . i by A4,A9;
      end; then
      w in product Carrier F by A5,CARD_3:9;
      hence ex w be Element of product F st P[z,w] by A3;
    end;
    consider I being Function of G, product F such that
    A10: for x being Element of G holds P[x,I.x] from FUNCT_2:sch 3(A6);
    reconsider I as Homomorphism of G, product F by Th19,A1,A10;
    I is bijective by Th20,A1,A10;
    hence thesis by A10;
  end;
