Why is $1 - \operatorname{P}(\text{Type I error}) =\overline{\operatorname{P}(\text{Type II error})}$?

Question

I understand that in hypothesis testing, an increase in $\operatorname{P}(\text{Type I error})$ will lead to a decrease in $\operatorname{P}(\text{Type II error})$. However, why does an addition of both probabilities not give 1?

As an extension, why then does $\text{Power} + \operatorname{P}(\text{Type II error}) = 1$?

Thank you in advance; your time is very much appreciated!