BG-MO 1975 Problem 2

Let $F$ be a polygon the boundary of which is a broken line with vertices in the knots (units) of a given in advance regular square network. If $k$ is the count of knots of the network situated over the boundary of $F$ , and $\ell$ is the count of the knots of the network lying inside $F$ , prove that if the surface of every square from the network is $1$ , then the surface $S$ of $F$ is calculated with the formulae: $$S=\frac k2+\ell-1$$ V. Chukanov