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 Th70:
  for G being finite Group
  for K,N being strict normal Subgroup of G
  for m,d being Nat
  st m = card N & m = card K & d = card(K /\ N)
  holds d*card(N "\/" K) = m*m
proof
  let G be finite Group;
  let K,N be strict normal Subgroup of G;
  let m,d be Nat;
  assume A1: m = card N;
  assume A2: m = card K;
  assume A3: d = card(K /\ N);
  reconsider B=K as Subgroup of G;
  A4: N is Subgroup of B "\/" N by GROUP_4:60;
  (B "\/" N)./.(B "\/" N,N)`*`, B./.(B /\ N) are_isomorphic by GROUP_6:81;
  then d*card(B "\/" N) = card(B) * card((B "\/" N,N)`*`) by A3,Th69
                       .= card(B)*card(N) by A4,GROUP_6:def 1
                       .= card(B)*m by A1
                       .= m*m by A2;
  hence d*card(N "\/" K) = m*m;
end;
