**Proof:**


((:) Conversely, let for every element of *N*<sup>p</sup> all partial conjugates of that element lie in Np: Then it comes directly from partial subgroup definition Conclusion(s). It is preferable to include a Conclusion(s) section which will summarize the content of the book chapter.

**Remark 3.1.** Any partial subgroup *H*p⊆ *G*<sup>p</sup> has right and left congruence (equivalence) class that cannot be the same. But if left and right congruence classes are the same (i.e., for any *x*∈ *G*p, *H*p.*x* = *x*.*H*p) then *H*<sup>p</sup> is called as normal partial subgroup.

**Theorem 3.2.** Let *G*<sup>p</sup> be a partial group and *N*<sup>p</sup> be a partial subgroup of a partial group *G*<sup>p</sup> and so the following conditions are coincided:

**Proposition 3.1.**

i. *N* <sup>p</sup> is a partial normal subgroup of a partial group *G*p.

ii. gp.*N*p=*N*p.gp for all gp∈ *G*p,

iii. gp.*N*p.gp �1⊆ *N*<sup>p</sup> for all gp∈ *G*p.
