Dynamic Fuzzy Machine Learning

The prior of (5) is proved. The after is similar. Having

$\begin{array}{l}Having\left(\overleftarrow{a},\overrightarrow{a}\right)\le \left(\overleftarrow{a},\overrightarrow{a}\right),\left(\overleftarrow{a},\overrightarrow{a}\right)\le \left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overrightarrow{b}\right),\text{then}\\ \left(\overleftarrow{a},\overrightarrow{a}\right)\le \left(\overleftarrow{a},\overrightarrow{a}\right)\Lambda \left(\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overrightarrow{b}\right)\right)\le \left(\overleftarrow{a},\overrightarrow{a}\right);\text{thus},\\ \left(\overleftarrow{a},\overrightarrow{a}\right)\Lambda \left(\left(\overleftarrow{a},\overrightarrow{a}\right)\vee \left(\overleftarrow{b},\overrightarrow{b}\right)\right)=\left(\overleftarrow{a},\overrightarrow{a}\right).\\ \text{Corollaries:}\left(\overleftarrow{a},\overrightarrow{a}\right)\Lambda \left(\left(\overleftarrow{a},\overrightarrow{a}\right)\Lambda \left(\overleftarrow{b},\overrightarrow{b}\right)\right)=\left(\overleftarrow{a},\overrightarrow{a}\right)\end{array}$

Definition 8.22 We call partial order set $\left(\left(\overleftarrow{L},\overrightarrow{L}\right),\le \right)$ semilattice; when $\forall \left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right),$$\left(\overleftarrow{\beta },\overrightarrow{\beta }\right)\in \left(\overleftarrow{L},\overrightarrow{L}\right),\text{we}\text{have}\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right)\vee (\overleftarrow{\beta }$