New theorem about core part of funcoids and reloids

Today I’ve proved a new little theorem:

Theorem $\mathrm{Cor} ( \mathsf{FCD}) g = ( \mathsf{FCD}) \mathrm{Cor}\, g$ for every reloid $g$.

Conjecture For every funcoid $g$

1. $\mathrm{Cor} ( \mathsf{RLD})_{\mathrm{in}} g = ( \mathsf{RLD})_{\mathrm{in}} \mathrm{Cor}\, g$;
2. $\mathrm{Cor} ( \mathsf{RLD})_{\mathrm{out}} g = ( \mathsf{RLD})_{\mathrm{out}} \mathrm{Cor}\, g$.

See my book.