Creating a Theorem with 4 Real Number Axioms and 2 Laws of Logic

[evagelos]

Given 4 axioms in real Nos:

1) 1x = x ,for all x

2) x+0 = x ,for all x

3) (x+y)z = xz+yz ,for all x,y,z

4) xy =yx ,for all x,y

The cancellation theorem : $$\forall[x+y = x+z\Longrightarrow y=z]$$

And two laws of logic:

1) The law of Universal Elimination

2) The law of substitution.

Can we create a theorem?.

If yes ,what that theorem may be??

 

[BWElbert]

Is this homework help or general inquiry? If homework help, what progress have you made before we answer? If simply general inquiry, I'll be more than happy to jump right in!

 

[evagelos]

Is this homework help or general inquiry? If homework help, what progress have you made before we answer? If simply general inquiry, I'll be more than happy to jump right in!

General inquiry.