 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);
reserve G1,G2 for Group;

theorem Th74:
  for G, A being finite Group
  for phi being Homomorphism of A,AutGroup(G)
  holds card semidirect_product(G,A,phi) = (card G)*(card A)
proof
  let G, A be finite Group;
  let phi be Homomorphism of A,AutGroup(G);
  set UG = the carrier of G;
  set UA = the carrier of A;
  A1: Carrier <* G, A *> = <* UG, UA *> by PRALG_1:18;
  card semidirect_product(G,A,phi) = card (product (Carrier <* G, A *>))
    by Def1
                                  .= card (product <* UG, UA *>) by A1
                                  .= (card UG)*(card UA) by GROUP_17:2;
  hence card semidirect_product(G,A,phi) = (card G)*(card A);
end;
