After learning individual rules and indirect proof, we now combine them in longer chains. Proofs using several rules of inference are written as sequences in which each line is either a premise or follows from earlier lines by a valid rule. This is the beginning of formal proof construction in discrete mathematics. In this lesson, proofs using several rules of inference will show how conditional conclusions, symbolic arguments, and verbal arguments are handled systematically.

Many arguments require more than one rule of inference. A proof of validity is written as a sequence of statements. Each statement is either a premise or follows from earlier statements by a valid rule of inference. The reason for every non-premise step should be stated so that the proof can be checked line by line.