reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;

theorem
  for N1,N2 be strict normal Subgroup of G ex N being strict normal
  Subgroup of G st the carrier of N = N1 * N2 & N1 ~ A \/ N2 ~ A c= N ~ A
proof
  let N1,N2 be strict normal Subgroup of G;
  consider N be strict normal Subgroup of G such that
A1:the carrier of N = N1 * N2 by Th8;
   N1 is Subgroup of N & N2 is Subgroup of N by A1,Th9; then
A2: N1 ~ A c= N ~ A & N2 ~ A c= N ~ A by Th26;
  N1 ~ A \/ N2 ~ A c= N ~ A
  proof
    let x be object;
    assume x in N1 ~ A \/ N2 ~ A;
    then x in N1 ~ A or x in N2 ~ A by XBOOLE_0:def 3;
    hence thesis by A2;
  end;
  hence thesis by A1;
end;
