This booklet bargains with the presentation and systematic layout of mathematical proofs, together with correctness proofs of algorithms. Its goal is to teach how completeness of argument, a big constraint in particular for the correctness of algorithms, may be mixed with brevity. the writer stresses that using formalism is indispensible for attaining this. A moment function of the e-book is to debate issues of layout. instead of addressing mental questions, the writer offers with extra technical questions like how research of the form of the demonstrandum can consultant the layout of an explanation. This technical instead of mental view of heuristics including the strain on exploiting formalism successfully are key gains of the e-book. The booklet comprises independently readable elements. One half features a variety of common chapters discussing options for transparent exposition, using formalism, the alternative of notations, the alternative of what to call and the way to call it, etc. the opposite half involves a chain of expositional essays, each one facing an explanation or an set of rules and illustrating using options mentioned within the extra normal chapters.