Hi guys! once again I'm sorry for posting like those question, but I really not succeeding solving those gaps by myself and thanks alot.

My problem is like this, lets assume that's x=y=z , then we conclude x=z ! it's really fine but it's a conclusion and not given data.

so now if I do z+m= 5, can I assign instead of z, x? I mean to write x+m=5 ?! what's confusing me because we conclude that x=z but it's not given information, given x=y=z, and from this we conclude x=z so I'm asking can I depend also on my implicitly conclusion of the given data(x=y=z) ?! thanks alot !

Your questions border on the incoherent.

You start with the GIVEN data that x = y = z. (Actually that is sloppy and confusing because equality is a binary relation. So what you really mean is "GIVEN x = y and y = z.")

Yes, you are

**ABSOLUTELY** correct that we are not told that x = z. Instead, we have an axiom that says

\(\displaystyle x = y \text { and } y = z \implies x = z.\)

It is not something that we must derive. It is an axiom that we are allowed to use without proof. It is a generalization of this example.

\(\displaystyle 3 + 8 = 11 \text { and } 11 = 17 - 6 \implies 3 + 8 = 17 - 6.\)

You can demonstrate physically the above example. Standard mathematics generalizes from that example and many similar examples to say that, in all cases,

\(\displaystyle x = y \text { and } y = z \implies x = z.\)

Now if you want, you can build a NON-STANDARD mathematics that denies that axiom, but I doubt it will be very useful when applied to the real world.