 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 Th61:
  for D being Subgroup-Family of F
  st (for i being Element of I holds D.i = (F.i)`)
  holds (product F)` is strict Subgroup of product D
proof
  let D be Subgroup-Family of F;
  assume A1: for i being Element of I holds D.i = (F.i)`;
  for a,b being Element of product F holds [. a, b .] in product D
  proof
    let a,b be Element of product F;
    B1: dom ([. a, b .]) = I by GROUP_19:3;
    for i being Element of I holds ([. a, b .]).i in D.i
    proof
      let i be Element of I;
      a/.i in (Omega).(F.i) & b/.i in (Omega).(F.i);
      then [. a/.i, b/.i .] in [. (Omega).(F.i),(Omega).(F.i) .] by GROUP_5:65;
      then [. a/.i, b/.i .] in (F.i)` by GROUP_5:def 9;
      then ([. a, b .]).i in (F.i)` by Th57;
      hence ([. a, b .]).i in D.i by A1;
    end;
    hence [. a, b .] in product D by B1,Th47;
  end;
  hence (product F)` is strict Subgroup of product D by GROUP_6:7;
end;
