Theorem 4.4

Theorem 4.4: If we compose two invertible isometries, the result is an invertible isometry.

Proof: Let M₁and M₂be the two isometries. Let A and B be two points in the plane. Then

|M₂(M₁(A)M₂(M₁(B)|

= |M₁(A)M₁(B)|

= |AB|

by the definition of an isometry.

Next, suppose that M₁and M₂are invertible, i.e., there exist M₁^-1 and M₂^-1 such that

M₁M₁^-1 = M₁^-1M₁ = I

and

M₂M₂^-1 = M₂^-1M₂ =

where I is the identity function. Then

(M₁M₂)(M₂^-1M₁^-1)

= M₁(M₂M₂^-1)M₁^-1

= M₁M₁^-1 = I

and

(M₂^-1M₁^-1)(M₁M₂)

= M₂^-1(M₁^-1M₁)M₂

= M₂^-1M₂ = I

So M₂^-1M₁^-1 is an inverse for M₁M₂.