Help me, I am new to discrete math

I was lost in the first day of class. He listed all the axiom from 1 to 13. Then he start proving things. For example:

Theorm
For all real number X
The additive inverse -X is unique.

Proof
Let Y and Z be additive inverse of X.

Y = Y + 0....................( axion 5, additive identity )
Y = Y + ( X + Z )..........( Z is additive inverse of X )
Y = (Y + X ) + Z...........( axion 3, associative law )
Y = 0 + Z.....................( Y is additive inverse of X )
Y = Z .........................( 0 is additive identity )

The prove above. I understand perfectly.

Here are some of the things that I need to prove but I don't know how to prove. Please teach/tell me how to do the following questions.

Question 3:
Prove that ( -X * Y ) is equal to ( X * -Y )

Question 6:
if X + Z = Y + Z prove that X equal to Y

Questions are from [link removed]

If you teach me how to do that those 2 question, I think i know how to do the rest hopefully.

Edited: To clarify, I was in a rush before

Based upon what you have posted, I surmise that the question deals with the axioms for the real number field. But from what you posted it is far from clear what the question happens to be. Why don’t you put up a definite and clearly stated question?

I edited my post. Please help me.

This attached file may help you.

Wow, this pdf is almost exactly what the teacher taught the first day. I rather look at this pdf then listen to my teacher, since I don't understand him.

There are some thing that i need to ask. For those theorem, do we need to memorize them? Because when i look at theorem 6, the person seem to refer it all to the previous theorem.