 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);

theorem Th60:
  for G being Group
  for H being Subgroup of G
  for N being strict normal Subgroup of G
  for phi being Homomorphism of H,AutGroup N
  ex psi being Function of semidirect_product (N,H,phi),G
  st (for n,h being Element of G st n in N & h in H
      holds psi.(<*n,h*>) = n*h)
  & (psi is one-to-one iff N /\ H = (1).G)
proof
  let G be Group;
  let H be Subgroup of G;
  let N be strict normal Subgroup of G;
  let phi be Homomorphism of H, AutGroup N;
  A1: for f being Function of product Carrier <*N,H*>,G
  holds f is Function of semidirect_product (N,H,phi), G
  proof
    let f be Function of product Carrier <*N,H*>,G;
    the carrier of semidirect_product (N,H,phi) = product Carrier <*N,H*>
    by Def1;
    hence f is Function of semidirect_product (N,H,phi), G;
  end;
  consider psi being Function of product Carrier <*N,H*>,G such that
  A2: (for n,h being Element of G st n in N & h in H holds psi.(<*n,h*>)=n*h)
  & (psi is one-to-one iff N /\ H = (1).G) by Th59;
  reconsider psi as Function of semidirect_product (N,H,phi), G by A1;
  take psi;
  thus thesis by A2;
end;
