# Two conjectures proved

In this online article I’ve proved:

Theorem $\mathrm{dom}\, (\mathsf{RLD})_{\mathrm{in}} f = \mathrm{dom}\, f$ and $\mathrm{im}\, (\mathsf{RLD})_{\mathrm{in}} f = \mathrm{im}\, f$ for every funcoid $f$.

and its easy consequence:

Proposition $\mathrm{dom}\, (\mathsf{RLD})_{\Gamma} f = \mathrm{dom}\, f$ and $\mathrm{im}\, (\mathsf{RLD})_{\Gamma} f = \mathrm{im}\, f$ for every funcoid $f$.