I'm writing up my solutions to a rather large set of number theory problems, and was wondering the following.
I'm certainly used to writing formal, 'structured' mathematics solutions, but in these problems I've frequently found I want to split my answers into several lemmas that together lead to a main theorem. This has prompted me to start using formal 'lemma, theorem, proof' formatting which I've never done before.
However, the markers want to see 'how I arrived at' my solutions as they are mainly interested in my thought process. For this reason I've been writing in normal prose, describing my thinking, and arriving every now and then at a main lemma or theorem.
My question is, is it always necessary to then include a formal proof of the theorem after its statement, if I've already explained how I got there? I've found that doing so leads to clumsy repetition (often using the same variables) in a slightly different order of the reasoning that lead me to the theorem - because I know proofs should work forwards from your assumptions, whereas my reasoning often works backwards from the result to work out how to get there.
I am aware that it is good practice to include formal proofs but if the proof is implied in my explanation leading up to the theorem, is it still necessary to include it formally?