The question is: Prove 3.4 from 3.2.

3.4 : If M1 and M2 are distinct planes, and each is parallel to plane M3, then M1 is parallel to M2

3.2 : Given plane M and point P not on M, there exists a unique plane through P parallel to M.

I use as an axiom the fact that two parallel planes do not intersect.
Here is the outline of how I try to write it...

There is a plane thru a point on M1 that is parallel to M2 ( by 3.2 )
This plane must equal M1 because any other plane thru that point would not be parallel to M3 (by 3.2)

and here it is...


Let M1, M2, and M3 be distinct planes such that M1 and M2 are both parallel to M3
Let P1, P2, and P3 be points on M1, M2, and M3 respectively
By 3.2 there exists a plane thru P1 that is parallel to M2. Call this plane M

If M was not equal to M1, then M would be not be parallel to M3 because by 3.2 there is only 1 plane that lies on P1 that is parallel to M3, and that plane is already defined to be M1.
If M was not parallel to M3, then M could not be parallel to M2 because M2 is parallel to M3.
This is impossible because M was defined to be parallel to M2.
Therefore, M must be M1 (if it is not M1 then it is not parallel to M2).

My questions are: did I prove 3.4; do I make any mistakes; how can I write this better?

Thanks I am terrible at writing these and I appreciate any help.