Composite with Constant Mapping is Constant Mapping

Theorem
Let $f_c: S \to T$ be the constant mapping defined as:
 * $\forall x \in S: \map {f_c} x = c$

where $c \in T$.

Then for all mappings $g: \Dom g \to S$:
 * $f_c \circ g$ is a constant mapping

and for all mappings $h: T \to \Cdm h$:
 * $h \circ f_c$ is a constant mapping

where:
 * $\Dom g$ denotes the domain of $g$
 * $\Cdm h$ denotes the codomain of $h$
 * $\circ$ denotes composition of mappings.

Proof
As $c$ is constant, so is $\map h c$.