Indirect proofs and proofs from assumptions

By a proof by assumption, or for short an assumption proof, I mean the following. The proof proves a conclusion of the form ‘If P then Q’. It proceeds by making a sequence of statements S1, . . . , Sn (possibly with added commentary), where P is S1, Q is Sn, and for each Si with i > 1 there are earlier Sj . . . such that the conditional ‘If Sj . . . , then Si’ is already known. The reason for calling these arguments ‘proofs by assumption’ is that they often begin with a phrase ‘Assume’ or ‘Suppose’ or ‘Let’, though one does find examples without this decoration. There are also quite a lot of examples where the assumptions consist of S1 and S2 together, so that the statement proved is ‘If S1 and S2 then Q’. I ignore this refinement in what follows. The statement being proved quite often takes one of the forms