Two theorems about totally bounded images of a totally bounded reloid

I’ve added to my book two following theorems (formerly conjectures).

Theorem Let $latex \mu$ and $latex \nu$ are endoreloids. Let $latex f$ is a principal $latex \mathrm{C}’ ( \mu; \nu)$ continuous, monovalued, surjective reloid. Then if $latex \mu$ is $latex \beta$-totally bounded then $latex \nu$ is also $latex \beta$-totally bounded.

Theorem Let $latex \mu$ and $latex \nu$ are endoreloids. Let $latex f$ is a principal $latex \mathrm{C}” (\mu ; \nu)$ continuous, surjective reloid. Then if $latex \mu$ is $latex \alpha$-totally bounded then $latex \nu$ is also $latex \alpha$-totally bounded.