theorem Th117:
  H is normal Subgroup of G iff for a holds a + H = H + a
proof
  thus H is normal Subgroup of G implies for a holds a + H = H + a
  proof
    assume
A1: H is normal Subgroup of G;
    let a;
A2: carr(H * a) = (-a) + H + a by ThB59;
    carr(H * a) = the carrier of the addMagma of H by A1,Def13
      .= carr H;
    hence a + H = a + ((-a) + H) + a by A2,ThA33
      .= a + (-a) + H + a by ThB105
      .= 0_G + H + a by Def5
      .= H + a by Th37;
  end;
  assume
A3: for a holds a + H = H + a;
  H is normal
  proof
    let a;
    the carrier of H * a = (-a) + H + a by ThB59
      .= H + (-a) + a by A3
      .= H + ((-a) + a) by ThA107
      .= H + 0_G by Def5
      .= the carrier of H by ThB109;
    hence thesis by Th59;
  end;
  hence thesis;
end;
