
theorem Th27:
  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 H,(product F0) st
  not q in I0 & I = I0 \/ {q} & F = F0 +* (q .--> K) & G0 is bijective
  ex G be Homomorphism of product <*H,K*>,(product F) st
  G is bijective &
  for h be Element of H,k be Element of K
  ex g be Function st g=G0.h & G.(<*h,k*>) = g +* (q .--> 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 H,(product F0);
    assume A1: not q in I0 & I = I0 \/ {q} &
    F = F0 +* (q .--> K) & G0 is bijective;
    set HK = <*H,K*>;
    A2: the carrier of product F0 = product Carrier F0 by GROUP_7:def 2;
    defpred P[set, set] means
    ex h be Element of H,k be Element of K,
    g be Function st $1 = <*h,k*> & g =G0.h & $2 = g +* (q .--> k);
    A3:for z be Element of product <*H,K*>
    ex w be Element of the carrier of (product F) st P[z,w]
    proof
      let z be Element of product <*H,K*>;
      consider h be Element of H,k be Element of K
      such that A4: z = <*h,k*> by TOPALG_4:1;
      consider g be Function such that
      A5: G0.h = g & dom g = dom (Carrier F0) &
      for y be object st y in dom (Carrier F0)
      holds g.y in (Carrier F0).y by CARD_3:def 5,A2;
      set w = g +* (q .--> k);
      w in the carrier of (product F) by A1,A5,Th22;
      hence thesis by A4,A5;
    end;
    consider G being Function of product <*H,K*>, product F such that
    A6: for x being Element of product <*H,K*>
    holds P[x,G.x] from FUNCT_2:sch 3(A3);
    A7:for h be Element of H,k be Element of K
    holds ex g be Function
    st g=G0.h & G.(<*h,k*>) = g +* (q .--> k)
    proof
      let h be Element of H,k be Element of K;
      reconsider z= <*h,k*> as Element of product <*H,K*>;
      consider h1 be Element of H,k1 be Element of K,
      g be Function such that
      A8: z = <*h1,k1*> & g =G0.h1 & G.z = g +* (q .--> k1) by A6;
      A9: h1 =(<*h1,k1*>).1
      .= h by FINSEQ_1:44,A8;
      k1 =(<*h1,k1*>).2
      .= k by FINSEQ_1:44,A8;
      hence thesis by A8,A9;
    end;
    then reconsider G as Homomorphism of product <*H,K*>,(product F)
    by Th23,A1;
    G is bijective by Th24,A1,A7;
    hence thesis by A7;
  end;
