theorem Th13:
  for x being object holds
  for N being normal Subgroup of G holds x in Cosets N implies
  ex a st x = a * N & x = N * a
proof let x be object;
  let N be normal Subgroup of G;
  assume x in Cosets N;
  then consider a such that
A1: x = a * N by GROUP_2:def 15;
  x = N * a by A1,GROUP_3:117;
  hence thesis by A1;
end;
