reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;
reserve K for characteristic Subgroup of G;

theorem Th69:
  for G1,G2 being finite Group
  for N1 being normal Subgroup of G1
  for N2 being normal Subgroup of G2
  st G1./.N1, G2./.N2 are_isomorphic
  holds card(N2)*card(G1)=card(N1)*card(G2)
proof
  let G1,G2 be finite Group;
  let N1 be normal Subgroup of G1;
  let N2 be normal Subgroup of G2;
  assume G1./.N1, G2./.N2 are_isomorphic;
  then A1: card(G1./.N1) = card(G2./.N2) by GROUP_6:73
                        .= index N2 by GROUP_6:27;
  set k = index N1;
  A2: card(G1) = card(N1) * index(N1) by GROUP_2:147
              .= card(N1) * k;
  card(G2) = card(N2) * index(N2) by GROUP_2:147
          .= card(N2) * k by A1,GROUP_6:27;
  hence thesis by A2;
end;
