Chapter 4

Method for Demonstration

4.1 Basic Method for Demonstration

We will explain the demonstration method. For a demonstration method, as we have explained before, we will first write a formula that needs to be demonstrated and then we will write the demonstration in between proof and end. 



    A formula that is needs to be demonstrated



     proof



      ...



      ...



      thus (hence) ...;



     end;

The end of a demonstration will always finish with thus or hence.

Let us demonstrate a proposition p that implies q.



       p implies q



        proof



         assume p;              p is assumed



          ...
 


          ...



         thus q;                q could be indicated



        end;

First, we will initially write a formula p that implies q that needs to be demonstrated, and write proof , end;. We do not need a semicolon after proof; however, there has to be one after end. If it is p, we need to demonstrate that it is q; however, p is also one logical formula. We will first write assume p and then p is assumed, and if we can indicate q, the demonstration is finished with thus q.

Next, if we want to demonstrate b=c that implies a=b after a=c is assumed, the following can be stated. With assume a=c, we will place label A:. 



   A:a=c;



    b=c implies a=b



    proof



     assume B:b=c



     hence thesis by A;



    end;

With assume ..., we hypothesize ... . In short, we indicate that it is a hypothesis section. Also, we should indicate that for hence ..., the last formula of this demonstration could be derived from a formula just before hence and the reasoning section of a hence formula. A thesis means that a formula should be indicated.