Prop. 3.1 (i)

[Brandon Batchelor]

Prop. 3.1: For any two points A and B:
i. ->AB and ->BA = AB
ii.->AB or ->BA = <->AB
Proof:
i. Let C be a point in AB. Therefore C is in the set {{C: A*C*B}U{A}U
{B}}.
Note that C is in the set ->AB (defn of ray ->AB).
C is also in the set ->BA (defn of ray ->BA).
Therefore C is in ->AB and ->BA.

Now let C be a point in ->AB and ->BA.
If C in ->AB then one of the following holds:
C = A or C = B => C is in AB. (defn of ray ->AB)
A*C*B => C is in AB. (B-1)
C*A*B => C is not in ->AB which violates earlier assumption.
(B-3)
A*B*C => C is not in ->BA which violates earlier assumption.
(B-3)

If C in ->BA then one of the following holds:
C = A or C = B => C is in AB. (defn of ray ->BA)
B*C*A => C is in AB. (B-1)
C*A*B => C is not in ->AB which violates earlier assumption.
(B-3)
A*B*C => C is not in ->BA which violates earlier assumption.
(B-3)

Since it was shown if C in ->AB and ->BA => C in AB and if C in AB
=> C in ->AB and ->BA then
i. ->AB and ->BA = AB holds.