Suppose that a maximal test performed at the time t_{1} yields the outcome *u*, and that we want to calculate the probability with which a maximal test performed at the later time t_{2} yields the outcome *w*. Further suppose that at some intermediate time t another maximal test is made, and that its possible outcomes are *v*_{1}, *v*_{2}, *v*_{3},… Let *v* be one of these values. Because a maximal test renders the outcomes of earlier measurements irrelevant, the joint probability p(*w*,*v*|*u*) with which the intermediate and final tests yield *v* and *w*, respectively, given the initial outcome *u*, is the product of two probabilities: the probability p(*v*|*u*) of *v* given *u*, and the probability p(*w*|*v*) of *w* given *v*. By Born’s rule, this is

p(*w*,*v*|*u*) = |<**w**|**v**> <**v**|**u**>|^{2},

where **u**, **v**, and **w** are unit vectors in the subspaces representing *u*, *v*, and *w*, respectively. To obtain the probability of *w* given *u*, *regardless* of the intermediate outcome, we must calculate this probability for all possible intermediate outcomes and add the results:

p_{A}(*w*|*u*) = |<**w**|**v**_{1}> <**v**_{1}|**u**>|^{2} + |<**w**|**v**_{2}> <**v**_{2}|**u**>|^{2} + |<**w**|**v**_{3}> <**v**_{3}|**u**>|^{2} + ···

In (other) words, first square the magnitudes of the amplitudes <**w**|**v**_{1}> <**v**_{1}|**u**>, etc., then add the results. This is Rule A. (Hence the subscript A.)

Rule B next. Since the vectors **v**_{1}, **v**_{2}, **v**_{3},… form a basis, any vector **u** can be written as

**u** = u_{1}**v**_{1} + u_{2}**v**_{2} + u_{3}**v**_{3} + ···,

where u_{1}, u_{2}, u_{3}, … are the components of **u** with respect to this basis. The scalar product of **v**_{1} and **u** is

<**v**_{1}|**u**> = u_{1}<**v**_{1}|**v**_{1}> + u_{2}<**v**_{1}|**v**_{2}> + ···.

Since basis vectors are unit vectors, we have that <**v**_{1}|**v**_{1}> = 1, and since they are mutually orthogonal, we have that <**v**_{1}|**v**_{2}> = 0. Hence <**v**_{1}|**u**> = u_{1}, <**v**_{2}|**u**> = u_{2}, etc. Consequently,

**u** = **v**_{1} <**v**_{1}|**u**> + **v**_{2} <**v**_{2}|**u**> + **v**_{3} <**v**_{3}|**u**> + ···

and

<**w**|**u**> = <**w**|**v**_{1}> <**v**_{1}|**u**> + <**w**|**v**_{2}> <**v**_{2}|**u**> + <**w**|**v**_{3}> <**v**_{3}|**u**> + ···.

If no intermediate measurement is made, the probability of *w* given *u* is

p_{B}(*w*|*u*) = |<**w**|**u**>|^{2} = |<**w**|**v**_{1}> <**v**_{1}|**u**> + <**w**|**v**_{2}> <**v**_{2}|**u**> + <**w**|**v**_{3}> <**v**_{3}|**u**> + ···|^{2}

In (other) words, first add the amplitudes <**w**|**v**_{1}> <**v**_{1}|**u**>, etc., then square the magnitude of the result. This is Rule B. (Hence the subscript B.)