 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem
  for I being finite non empty set
  for F being Group-Family of I
  for D being Subgroup-Family of F
  st (for i being Element of I holds D.i = (F.i)`)
  holds (product F)` = product D
proof
  let I be finite non empty set;
  let F be Group-Family of I;
  let D be Subgroup-Family of F;
  assume A1: for i being Element of I holds D.i = (F.i)`;
  sum D = product D by GROUP_7:9;
  then A2: product D is strict Subgroup of (product F)` by A1, Th62;
  (product F)` is strict Subgroup of product D by A1, Th61;
  hence thesis by A2, GROUP_2:55;
end;
