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 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 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 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.

Let's explain thus and hence a little in more detail.

Although thus and hence are used in proof, the formula which
should be proved becomes easy every time hence and thus appear and a proved
part is removed from the formula connected by &. For example, the following
is the proof of three formulas connected by and.

theorem 3-1=2 & 5-2=3 & 6-3=3

proof 2+1=3;

hence 3-1=2 by REAL_2:17;

3+2=5;

hence 5-2=3 by REAL_2:17;

3+3=6;

hence thesis by REAL_2:17;

end;

Since 3-1=2 are proved by the first hence, we become the formula which
5-2=3 & 6-3=3 should prove. Since 5-2=3 are proved by the next hence,
6-3=3 become the formula which should finally be proved. It means that
all the remaining parts (6-3=3) are first proved by hence thesis.

In addition, hence combines then and thus, as mentioned above.

A1: A=B;

A2: B=C;

thus A=C by A1,A2;

For example, we can write the above as the following.

A1: A=B;

B=C;

hence A=C by A1;