 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 Th43:
  for g being Element of G
  holds g in Ker (product f)
        iff (for i being Element of I holds g in Ker (f.i))
proof
  let g be Element of G;
  thus g in Ker (product f)
       implies (for i being Element of I holds g in Ker (f.i))
  proof
    assume A1: g in Ker (product f);
    let i be Element of I;
    (product f).g = 1_(product F) by A1, GROUP_6:41;
    then 1_(F.i) = ((product f).g).i by Th42
                .= (f.i).g by Def14;
    hence g in Ker (f.i) by GROUP_6:41;
  end;
  thus (for i being Element of I holds g in Ker (f.i))
       implies g in Ker (product f)
  proof
    assume A1: for i being Element of I holds g in Ker (f.i);
    A2: for i being Element of I holds (f.i).g = 1_(F.i)
    proof
      let i be Element of I;
      g in Ker (f.i) by A1;
      hence (f.i).g = 1_(F.i) by GROUP_6:41;
    end;
    for i being Element of I holds ((product f).g).i = 1_(F.i)
    proof
      let i be Element of I;
      ((product f).g).i = (f.i).g by Def14
                       .= 1_(F.i) by A2;
      hence thesis;
    end;
    then (product f).g = 1_(product F) by Th42;
    hence g in Ker (product f) by GROUP_6:41;
  end;
end;
