Equivalence of Definitions of Matroid Rank Axioms/Lemma 3

Theorem
Let $S$ be a finite set.

Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.

Let $\rho$ satisfy the rank axioms:

Let:
 * $A \subseteq S : \map \rho A = \card A$

Let:
 * $B \subseteq S : \forall b \in B \setminus A : \map \rho {A \cup \set b} \ne \card{A \cup \set b}$

Then:
 * $\map \rho {A \cup B} = \map \rho A$