
theorem Th28:
  for I0,I be non empty finite set,
  F0 be associative Group-like multMagma-Family of I0,
  F be associative Group-like multMagma-Family of I,
  H,K be Group,
  q be Element of I,
  G0 be Homomorphism of product F0, H st not q in I0 &
  I = I0 \/ {q} & F = F0 +* (q .--> K) & G0 is bijective holds
  ex G be Homomorphism of product F, product <*H,K*> st G is bijective &
  for x0 be Function,
  k be Element of K,
  h be Element of H
  st h = G0.x0 & x0 in product F0 holds
  G.(x0 +* (q .-->k)) = <* h, k *>
  proof
    let I0,I be non empty finite set,
    F0 be associative Group-like multMagma-Family of I0,
    F be associative Group-like multMagma-Family of I,
    H,K be Group,
    q be Element of I,
    G0 be Homomorphism of product F0,H;
    assume A1: not q in I0 &
    I = I0 \/ {q} & F = F0 +* (q .--> K) & G0 is bijective;
    set L0=G0";
    A2: rng G0 = the carrier of H by FUNCT_2:def 3,A1; then
    reconsider L0 as Function of H, product F0 by FUNCT_2:25,A1;
    A3: L0*G0 = id the carrier of product F0
    & G0*L0 = id the carrier of H by A1,A2,FUNCT_2:29;
    A4:L0 is onto by A3,FUNCT_2:23;
    reconsider L0 as Homomorphism of H,product F0 by A1,GROUP_6:62;
    consider L be Homomorphism of product <*H,K*>,(product F) such that
    A5: L is bijective &
    for h be Element of H,k be Element of K
    holds ex g be Function
    st g=L0.h & L.(<*h,k*>) = g +* (q .--> k) by Th27,A1,A4;
    set G=L";
    A6: rng L = the carrier of (product F) by FUNCT_2:def 3,A5;
    then
    reconsider G as Function of product F, product <*H,K*> by FUNCT_2:25,A5;
    A7: G * L = id (the carrier of product <*H,K*>)
    & L * G = id (the carrier of (product F))
    by A5,A6,FUNCT_2:29;
    A8:G is onto by A7,FUNCT_2:23;
    reconsider G as Homomorphism of product F,(product <*H,K*>)
    by A5,GROUP_6:62;
    for x0 be Function, k be Element of K, h be Element of H
    st h = G0.x0 & x0 in (product F0) holds
    G.(x0 +* (q .-->k)) = <* h, k *>
    proof
      let x0 be Function,
      k be Element of K,
      h be Element of H;
      assume A9: h = G0.x0 & x0 in (product F0);
      consider g be Function such that
      A10: g=L0.h & L.(<*h,k*>) = g +* (q .--> k) by A5;
      g = x0 by A10,A1,A9,FUNCT_2:26;
      hence G.(x0 +* (q .--> k)) = <*h,k*> by A5,FUNCT_2:26,A10;
    end;
    hence thesis by A8,A5;
  end;
