
theorem Th26:
  for q be set,
  F be associative Group-like multMagma-Family of {q},
  G be Group st F = q .--> G holds
  ex HFG be Homomorphism of product F,G st
  HFG is bijective &
  for x be (the carrier of G)-valued total {q} -defined Function
  holds HFG.x = Product x
  proof
    let q be set,
    F be associative Group-like multMagma-Family of {q},
    G be Group;
    assume A1: F = q .--> G;
    consider I be Homomorphism of G, product F such that
    A2: I is bijective &
    for x being Element of G holds I . x = q .--> x by Th21,A1;
    set HFG = I";
    A3: rng I = the carrier of (product F) by A2,FUNCT_2:def 3; then
    reconsider HFG as Function of product F, G by FUNCT_2:25,A2;
    A4: HFG*I = id (the carrier of G)
    & I*HFG = id (the carrier of (product F)) by A2,A3,FUNCT_2:29;
    A5:HFG is onto by A4,FUNCT_2:23;
    reconsider HFG as Homomorphism of product F,G by A2,GROUP_6:62;
    for y be (the carrier of G)-valued total {q} -defined Function
    holds HFG.y =Product y
    proof
      let y be (the carrier of G)-valued total {q} -defined Function;
      A6: y in the carrier of product F &
      y.q in the carrier of G &
      y= q .--> y.q by A1,Th25;
      reconsider z=y.q as Element of G by A1,Th25;
      A7: I . z = q .--> z by A2
      .= y by A1,Th25;
      thus HFG.y = z by FUNCT_2:26,A2,A7
      .= Product y by Th9,A6;
    end;
    hence thesis by A5,A2;
  end;
