Given a partially ordered set $\left(P,\le\right)$ and a subset $S$ of $P$:
Theorem (I): If maximum $S$ does exist, it's unique.
Proof:
Assume there are two maximum of $S$ denoted $g_1,g_2$, then by definition of maximum it follows $g_{1}\le g_{2}$ and $g_{2}\le g_{1}$, using the anti-symmetric property of $S$ implies $g_{1}=g_{2}$.
Theorem (II): Every maximum is a maximal.
Proof:
A maximal is by definition greater than or equals to any other element that can be compared with , a maximum is by definition greater than or equals to any other element in $S$,clearly maximum is stronger than maximal and maximal is a generalization of maximum, hence the result follows.
Theorem (III) : If maximum of $S$ exists then the maximal of $S$ is unique.
Proof:
From definition of maximum it follows that maximum of $S$ denoted $g$ is greater than or equals to any other element $s$ in $S$, specially when $s$ is a maximal of $S$ , so $s\le g$, also since $s$ is the maximal of $S$ so by definition of maximal it follows that $s$ is greater than or equals to any other element in $S$ specially $g$ ,e.g. $g\le s$, concludes $g=s$ , since $g$ is unique by Theorem I) implies $s$ is also unique.
Since this is true for all partial-order relations and since partial-order relation is a generalization of a total-order relation hence in a total order relation maximum and maximal coincides.
Theorem (IV): The set of maximal elements of a subset S is always an anti-chain, that is, no two different maximal elements of S are comparable.
Proof:
Using theorem III) follows a subset $S$ of a partially ordered set $\left(P,\le\right)$ can have more than one maximal element if it does not have a greatest element, for the sake of contradiction assume there exist two maximal elements such that they are comparable, it means that they are related to each other via $\le$ and this contradicts the fact that the two elements are indeed maximal.
I'm not sure if all of the proofs are right, can someone please verify? and please let me know if there are other well-known theorems about this object.