Some question about lattice rank.

Question

I found this equation while looking at the "Rank of a partially ordered set"

"A lattice with a rank function ρ is (upper) semi-modular if:ρ(x)+ρ(y)≥ρ(x∨y)+ρ(x∧y)" (https://encyclopediaofmath.org/wiki/Rank_of_a_partially_ordered_set)

how to prove it?

the other question is how to prove:If x and y both cover x ∧ y, then x ∨ y covers both x and y?

Answer 1

There is this result in [1], page 226.

Theorem 2. Let $L$ be a lattice of finite length. The following conditions on $L$ are equivalent, for all $a,b,c \in L$:
(1) $L$ is semimodular.
(2) If $a\neq b$, $a$ and $b$ cover $a\wedge b$, then $a\vee b$ covers $a$ and $b$.
(3) If $a\leq b$ and $C$ is a maximal chain in $[a,b]$, then $\{x\vee c \mid x \in C\}$ is a maximal chain in $[a\vee c,b\vee c]$.
(4) $h(a)+h(b)\geq h(a\wedge b)+h(a\vee b)$.

Here, $h$ is the height function, which is the same as the rank, at least in the finite length case.
You are asking for the equivalences of (1), (2) and (4).

[1] Grätzer, G., General Lattice Theory, 2nd edition.