 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);

theorem Th52:
  for G being Group
  for N being normal Subgroup of G
  for g1,g2 being Element of G
  st g1 * N = g2 * N
  ex n being Element of G
  st n in N & g1 = g2 * n
proof
  let G be Group;
  let N be normal Subgroup of G;
  let g1,g2 be Element of G;
  assume A1: g1 * N = g2 * N;
  consider n being Element of G such that
  A2: n = (g2") * g1;
  take n;
  thus n in N by A1, A2, GROUP_2:114;
  thus g2 * n = (g2 * (g2")) * g1 by A2, GROUP_1:def 3
             .= (1_G) * g1 by GROUP_1:def 5
             .= g1 by GROUP_1:def 4;
end;
