Theorem 4.11: If   T   is the translation

T(x, y) = (x - x0, y - y0)

then the translation

U(x, y) = (x + x0, y + y0)

is the inverse of   T.

Proof: If we compose the two translations, we get

T(U(x, y))

= T(x + x0, y + y0)

=((x + x0) - x0, (y + y0) - y0 )

= (x, y)

If we compose them in the other order, we get

U(T(x, y))

= U(x - x0, y - y0)

=((x - x0) + x0, (y - y0) + y0 )

= (x, y)

so   U   acts like an inverse for   T.

next theorem (4.12)