reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;

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;
