 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
  the carrier of Ker (product f) =
    meet the set of all the carrier of Ker (f.i) where
      i is Element of I
proof
  set Fam = the set of all the carrier of Ker (f.i) where i is Element of I;
  A1: Fam <> {}
  proof
    set i = the Element of I;
    the carrier of Ker (f.i) in Fam;
    hence thesis;
  end;
  A3: for g being object
  holds g in Ker (product f)
        iff (for A being set st A in Fam holds g in A)
  proof
    let g be object;
    hereby
      assume B1: g in Ker (product f);
      let A be set;
      assume A in Fam;
      then consider i1 being Element of I such that
      B2: A = the carrier of Ker(f.i1);
      g in G & g in Ker (product f) by B1, GROUP_2:40;
      then g in Ker (f.i1) by Th43;
      hence g in A by B2;
    end;
    assume B1: for A being set st A in Fam holds g in A;
    g in G & for i being Element of I holds g in Ker (f.i)
    proof
      ex x being object st x in Fam by A1, XBOOLE_0:def 1;
      then consider A0 being set such that
      B2: A0 in Fam;
      consider i0 being Element of I such that
      B3: A0 = the carrier of Ker (f.i0) by B2;
      g in Ker (f.i0) by B1, B2, B3;
      hence g in G by GROUP_2:40;
      let i be Element of I;
      the carrier of Ker (f.i) in Fam;
      hence g in Ker (f.i) by B1;
    end;
    hence g in Ker (product f) by Th43;
  end;
  A4: for g being object holds g in Ker (product f) iff g in (meet Fam)
  proof
    let g be object;
    g in Ker (product f) iff (for A being set st A in Fam holds g in A) by A3;
    hence thesis by A1, SETFAM_1:def 1;
  end;
  for g being object
  holds g in the carrier of Ker (product f) iff g in meet Fam
  proof
    let g be object;
    g in the carrier of Ker (product f) iff g in Ker (product f);
    hence thesis by A4;
  end;
  hence thesis by TARSKI:2;
end;
