This formula (subject to proof or disproof in the theory) is a so-called “Hoare Triple”, {P} A {Q}, expressing that an execution of A started in a state satisfying P will yield a state satisfying Q. Here if we start with n greater than 5 we end up with n greater than 4. (And no, there is no typo in the example. The triple is not the strongest we can derive but it holds.) The way this works is that the theory consists of an axiom or inference rule for every construct of the language; for example the axiom for assignment (“:=”) is {P [E / n]} n := E {P}, working backwards, where P [E / n} denotes the assertion P in which every occurrence of n has been replaced by the expression E, textually (a concept that comes from lambda calculus, designed by Church and Curry in the 1930s, and works whether or not E includes occurrences of n, as it does in the above n := n + m example).
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