Theorem 4.4: If we compose two invertible isometries, the result is an invertible isometry.
Proof: Let M1 and M2 be the two isometries. Let A and B be two points in the plane. Then
by the definition of an isometry.
Next, suppose that M1 and M2 are invertible, i.e., there exist M1-1 and M2-1 such that
M1M1-1 = M1-1M1 = I
M2M2-1 = M2-1M2 =I
where I is the identity function. Then
= M1M1-1 = I
= M2-1M2 = I
So M2-1M1-1 is an inverse for M1M2.
next theorem (4.5)