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 Th24:
  for N being normal Subgroup of G holds 1_(G./.N) = carr N
proof
  let N be normal Subgroup of G;
  reconsider e = carr N as Element of G./.N by GROUP_2:135;
  now
    let h be Element of G./.N;
    consider a such that
A1: h = a * N and
A2: h = N * a by Th13;
    thus h * e = (a * N) * N by A1,Def3
      .= a * (N * N) by GROUP_4:45
      .= h by A1,GROUP_2:76;
    thus e * h = N * (N * a) by A2,Def3
      .= N * N * a by GROUP_4:46
      .= h by A2,GROUP_2:76;
  end;
  hence thesis by GROUP_1:4;
end;
