 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 F being normal Subgroup-Family of I,G
  for A being Subset of G
  for i being Element of I
  st A = union { the carrier of F.j where j is Element of I : i <> j }
  ex N being strict normal Subgroup of G
  st N = gr A
proof
  let F be normal Subgroup-Family of I,G;
  let A be Subset of G;
  let i be Element of I;
  set X1 = {the carrier of F.j where j is Element of I : i <> j };
  assume A1: A = union X1;
  set J = I \ {i};
  per cases;
  suppose J is empty;
    then I <> {} & I c= {i} by XBOOLE_1:37;
    then A3: I = {i} by ZFMISC_1:33;
    not ex x being object st x in X1
    proof
      given x be object such that
A0:   x in X1;
      ex j being Element of I st x = the carrier of F.j & i <> j by A0;
      hence contradiction by A3, TARSKI:def 1;
    end;
    then X1 = {} by XBOOLE_0:def 1;
    then A4: A = {} the carrier of G by A1, ZFMISC_1:2, SUBSET_1:def 2;
    take N = (1).G;
    thus thesis by A4, GROUP_4:30;
  end;
  suppose J is non empty;
    then reconsider J as non empty set;
    reconsider FF=F|J as Subgroup-Family of J,G by ThSubFamRes;
    for j being Element of J holds FF.j is normal Subgroup of G
    proof
      let j be Element of J;
      j in I by XBOOLE_0:def 5;
      then F.j is normal Subgroup of G by ThS1;
      hence FF.j is normal Subgroup of G by FUNCT_1:49;
    end;
    then reconsider FF as normal Subgroup-Family of J,G by ThS1;
    set X2 = { the carrier of FF.j where j is Element of J :
      not contradiction };
    for x being object holds x in X1 iff x in X2
    proof
      let x be object;
      hereby
        assume x in X1;
        then consider j being Element of I such that
        Z1: x = the carrier of F.j & i <> j;
        j in I & not j in {i} by Z1, TARSKI:def 1;
        then reconsider jj=j as Element of J by XBOOLE_0:def 5;
        F.j = FF.jj by FUNCT_1:49;
        hence x in X2 by Z1;
      end;
      assume x in X2;
      then consider j being Element of J such that
      Z1: x = the carrier of FF.j;
      j in I & not j in {i} by XBOOLE_0:def 5;
      then Z2: j in I & j <> i by TARSKI:def 1;
      then reconsider ii=j as Element of I;
      the carrier of F.ii = x by Z1, FUNCT_1:49;
      hence x in X1 by Z2;
    end;
    then A3: X1 = X2 by TARSKI:2;
    then reconsider B = union X2 as Subset of G by A1;
    consider N being strict normal Subgroup of G such that
    A2: N = gr B by ThJoinNorm;
    take N;
    thus N = gr A by A1, A2, A3;
  end;
end;
