theorem
  for H being strict Subgroup of G holds H is normal Subgroup of G iff
  for a holds H * a is Subgroup of H
proof
  let H be strict Subgroup of G;
  thus H is normal Subgroup of G implies for a holds H * a is Subgroup of H
  proof
    assume
A1: H is normal Subgroup of G;
    let a;
    H * a = the addMagma of H by A1,Def13;
    hence thesis by ThA54;
  end;
  assume
A2: for a holds H * a is Subgroup of H;
  now
    let a;
A3: H * (-a) + a = -(-a) + H + (-a) + a by ThB59
      .= -(-a) + H + ((-a) + a) by ThB34
      .= -(-a) + H + 0_G by Def5
      .= a + H by Th37;
    H * (-a) is Subgroup of H by A2;
    hence a + H c= H + a by A3,ThB6;
  end;
  hence thesis by Th118;
end;
