theorem
  for N being normal Subgroup of G holds Left_Cosets N = Right_Cosets N
proof
  let N be normal Subgroup of G;
  thus Left_Cosets N c= Right_Cosets N
  proof
    let x be object;
    assume x in Left_Cosets N;
    then consider a such that
A1: x = a + N by Def15;
    x = N + a by A1,Th117;
    hence thesis by Def16;
  end;
  let x be object;
  assume x in Right_Cosets N;
  then consider a such that
A2: x = N + a by Def16;
  x = a + N by A2,Th117;
  hence thesis by Def15;
end;
