I would like to ask this question: -Do we have to present the demonstrations of the results (theorems in mathematics) found and published in a scientific journal the day of the PHD defense or it is enough to present the results only? Thank You.
That will depend on circumstances. Typically you will be examined by a committee of (mostly) professors in your field during your defense. The defense itself may be open to the public or not. Most of the committee will have read, or at least skimmed your dissertation.
Exceptions abound. Some committees have an outside member from a completely different field. Some are private with possibly a public talk before or afterwards. Some defenses are not just a defense of the thesis, but a defense of your entire understanding of the field.
You need to speak, primarily, to the committee. It may be that a proof is required. It may be that a proof is too long to give in such a talk. You should be able to, at least, give an outline of the important proofs and to say what, if anything, is novel about them.
In some mathematical work the theorems themselves are the most important thing. In others (quite a lot, actually) the proof methodology is more important. That is to say, the "what" vs the "how" of the thesis. Make sure you understand which it is (and if that is the case) and that you spend most of the time on that. You have to give a solid overview also, of course.
Get advice from your advisor on this. He or she will probably know your committee members and how they behave at a defense. Are they passive, are they active questioners, are they skeptical of everything? Are they generalists who want an overview or nit-pickers who want every detail?
If you have an outside examiner, be prepared to give reasonable "layperson" explanations of all of the basics that underlie your work. There is a story, perhaps apocryphal, of a candidate in Chemistry being failed after giving a good talk and writing a good thesis because she couldn't adequately explain Ph (acidity) to an outsider after using it extensively in the talk. She studied that so long ago that the simple explanation just escaped her. (This may be a bedtime horror story told to candidates just to scare them before their defense.)
On the other hand, some defenses are really just a celebration by the faculty of the wonderful work that they do in producing such fine candidates as yourself. Possibly ending with Jerez (Sherry).