@DevSinha Thanks for the information. In my case, unfortunately, they finally came to say, okay, all the rest of the world failed, so what, we can do it!!!

Could you please add a few lines for explanation.

Here is a review to the book I introduced. It is written by Jeremy Gray: www.ams.org/notices/200009/rev-gray.pdf

@quid I didn't down-vote. But I can imagine why. Just a 5 minutes search on the web will come up with an answer.

Here is one: "Zero: The Biography of a Dangerous Idea". Indeed, for your inquiry, it was just much easier to search the web than looking for an answer here.

@JoelReyesNoche I am sure MO users are faced with such down-to-earth situations in their everyday teaching life. I am happy that there are several occasions that MO has accepted that fact in the past. Let's hope for the future.

@HenryCohn Indeed, as I mentioned before, he is a highly competent mathematics student. I asked him why he asked such a question. Here is the story. He was teaching a high school math course when his students came up with the question. His answer was "as far as I know, in mathematics a proof is constructed based on axioms". I feel, his answer works for him (us) individually, but it is not a constructive answer for his (our) students as such. That is why I asked the question.

@TheoJohnson-Freyd It is indeed a very constructive modification. Thanks. Consequently I need to modify my description. I'll do it now.

The first way has a more educational flavor that is going better with the flow of thinking. It is like saying that "now, it is enough to show that..."; and that is your lemma.