I'm reading Bede's Mathematics of Fuzzy Sets and Fuzzy Logic and encountered the following proof for the absorption law:
where $\land$, $\lor$ and $\leq$ stand for "intersect", "union", and "include".
It is clear that $A(x) \lor ( A(x) \land B(x)) \leq A(x) \lor A(x)$ holds but I do not understand how it is transformed to $$ A(x) \leq A(x) \lor( A(x) \land B(x))$$
How to justify that replacing $A(x)$ with $(A(x) \land B(x))$ at the right hand side of $\leq$ cancels $\lor (A(x) \land B(x))$ at the left hand side? Or how should I interpret the transformation?
Oh. It makes more sense if the expression is viewed in a single-line:
$$A\lor (A \land B)(x) = A(x) \lor ( A(x) \land B(x)) \leq A(x) \lor A(x) = A(x) \leq A(x) \lor( A(x) \land B(x))$$
where the second equal sign denotes the equality between $A(x) \lor A(x) = A(x)$ which in turns is included in $A(x) \lor( A(x) \land B(x))$, thus proving that
$$A\lor (A \land B)(x) \leq A(x) \leq A\lor (A \land B)(x)$$
i.e.
$$A\lor (A \land B)(x) = A(x)$$