theorem
  for N being normal Subgroup of G holds Left_Cosets N is finite implies
  card(G./.N) = index N
proof
  let N be normal Subgroup of G;
  assume Left_Cosets N is finite;
  then reconsider LC = Left_Cosets N as finite set;
  thus card(G./.N) = card LC .= index N by GROUP_2:def 18;
end;
