theorem Th118:
  for H being Subgroup of G holds H is normal Subgroup of G iff
  for a holds a + H c= H + a
proof
  let H be Subgroup of G;
  thus H is normal Subgroup of G implies for a holds a + H c= H + a by Th117;
  assume
A1: for a holds a + H c= H + a;
  now
    let a;
    (-a) + H c= H + (-a) by A1;
    then a + ((-a) + H) c= a + (H + (-a)) by Th4;
    then a + (-a) + H c= a + (H + (-a)) by ThB105;
    then 0_G + H c= a + (H + (-a)) by Def5;
    then carr H c= a + (H + (-a)) by ThB109;
    then carr H c= a + H + (-a) by ThB106;
    then carr H + a c= a + H + (-a) + a by Th4;
    then H + a c= a + H + ((-a) + a) by ThB34;
    then H + a c= a + H + 0_G by Def5;
    hence H + a c= a + H by Th37;
  end;
  then for a holds a + H = H + a by A1;
  hence thesis by Th117;
end;
