This example is taken from the elementary number theory of the natural numbers. It contains just one definition and one theorem. The definition is that of divisibility. The big V is an existence quantifier. The definition states that a divides b if and only if there is a natural number k which multiplied by a yields b. The theorem states that divisibility is transitive.
Notes 1 and 2 of the proof state the hypotheses of the theorem as givens. Givens are justified with a G. Note 3 is an assignment statement. Assignment statements are justified with an S. It sets k₁ to a natural number which multiplied by a yields b. Similarly in Note 4 we pick a k₂ which multiplied by b yields c. In Note 5 we let k be the product of k₁ and k₂ and in Notes 6 and 7 we verify that k is also a natural number. The proof of Note 6 appeals to Definition 1.1 and to Note 1. Checking Note 6 requires a reference to Theorem 40.44 in the Common Notions file which says that the product of two natural numbers is a natural number.
Note 9 then uses the associativity of multiplication and the k defined in Note 5 to get c equal to k times a. The reference to the associativity of multiplication is 039.7 and refers to Theorem 39.7 of in the common notions file. The H followed by .1 and .2 in the QED reference forms an inference from the two givens to Note 9, which constitutes the theorem to be proven.