Why is the composition of a liquified vapor the same as that of the vapor?
There is a misconception in using this diagram. You cannot go from Point a to Point b This diagram should not be read as you do. It should be done this way.

Suppose you start from a liquid made of 25% A + 75% B. It is a mixture. Its representative point on the diagram is on the edge of the rectangle between "Composition" and "100% B". When heating this mixture, the corresponding point moves upwards, but stays liquid. When reaching the curve "Liquid", the first bubble of gas is formed. We are at the point a (red) But this bubble has another composition. The composition of the gas corresponds to b (in red). So it is practically pure A. This is the beginning of the vaporization phenomena. 

If you continue heating the liquid, more gas will be produced. But, as A is more eliminated than B by vaporization, the remaining liquid looses some A and practically no B. So in percentage, the liquid will contain more B. Its composition will soon be 76% B and 24% A, then 77% B and 23 % A, etc. But the temperature has to be elevated to allow this richer mixture to vaporize. The representative point of the liquid will move but stay on the sloping curve. After some time, the point of the liquid will be just above the capital L from the word "Liquid" At this moment the gas is mainly A but it starts containing 1 or 2% B. The composition of the gas could be determined by drawing a horizontal line from the point above the letter "L". This horizontal line will cut the curve "Gas" at an abscissa equal to about 1% or 2%. This defines the composition of the gas. 

If you continue heating, the liquid will contain more and more B. Its point will stay on the curve "Liquid". The temperature must be higher and higher. The amount of liquid decreases. Simultaneously the composition of the gas which is eliminated contains more and more B. Its composition is always determined by drawing a horizontal line from the liquid to the curve on the left-side of the diagram. 

The last drop of liquid disappears when the liquid reaches the Point "bp B". 

Hopefully you have understood that passing from point a (in red) to point b (in red) has no meaning. a and b are always present : one describes the composition of the liquid, and one the composition of the gas.