– (6,4); (5.5,3.5) – (6,3.5);

(7,4.5) – (7.5,4.5); (7.5,4.5) – (7.5,5); (7.7, 5.2) – (8.2,5.4);

In Figure \[boardB+\] we have drawn $P'$ up and to the right of $P$. It is also possible that $P'$ is down and to the left of $P$. In this case the same proof will work by using the transpose of $B^+$.

Let $$A_{{\overline}{S}}: = \{M\in M(B^+): P'\in M, P \notin M, S\cap M=\emptyset\},$$ $$A_{{\overline}{T}}: = \{M\in M(B^+): P\in M, P'\notin M T\cap M=\emptyset\},$$ $$A_{S}: = \{M\in M(B^+): P'\in M, P\notin M S\cap M\neq\emptyset\},$$ $$A_{T}: = \{M\in M(B^+): P\in M, P'\notin M, T\cap M\neq\emptyset\},$$ We first define a bijection between $A_{{\overline}{S}}$ and $A_{{\overline}{T}}$ and then we define an injection from $A_S$ to $A_T$. For each case, we will define a replacement function $r_i$ on the blocks of $B^+$ and then an injection $f_i$ on the appropriate matchings. In the end we will have $$A_{{\overline}{S}}\overset{f_1}{\longleftrightarrow}A_{{\overline}{T}}$$ $$A_{S} \overset{f_2}{\hookrightarrow} A_T.$$ Since each map will send matchings of size $k$ to matchings of size $k$, we conclude $m_k(B')\leq m_k(B)$ for all $k \geq 0$.

*Case 1:* Suppose $M\in A_{{\overline}{S}}$, that is, $M$ is a matching in $B^+$ such that $P'\in M$, $P\notin M$, and $S\cap M =\emptyset$. Define $r_1: 