Equivalence of Definitions of Matroid Rank Axioms/Condition 2 Implies Condition 1

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 definition 2 of the rank axioms:

Then $\rho$ satisfies definition 1 of the rank axioms:

$\rho$ satisfies $(\text R 1)$
We have:

Hence:
 * $\map \rho \O = 0$

$\rho$ satisfies $(\text R 2)$
Let $X \subseteq S$.

Let $y \in S$.

We have:

This proves Rank axiom $(\text R 2)$

$\rho$ satisfies $(\text R 3)$
Let $X \subseteq S$.

Let $x, y \in S$.

Let $\map \rho {X \cup \set x} = \map \rho {X \cup \set y} = \map \rho X$.

We have:

Hence:
 * $\map \rho {X \cup \set x \cup \set y} = \map \rho X$

This proves Rank axiom $(\text R 3)$