What are provable by proper LCM-principles?

lWe** **are now classifying
mathematical theorems by means of these principles.
lThere are theorems exceeding the scope of LCM, like
Bolzano-Weierstrass theorem, but many theorems of standard
mathematics are in the scope of LCM
lE.g. Heine-Borel theorem, Completeness theorem of predicate
logic, etc.etc~{!-~}.