There is a theorem that if a set of edge colored square tiles can be used to tile the positive orthant then it can also be used to tile the entire plane. (The edges are colored and in the edge-to edge tiling you need to match the colors).
The easy argument (using infinity) is: draw a rooted tree where the lavel k level $k$ vertices are all possible tilings of a (2k-3) x $(2k-3) \times (2k-3) 2k-3)$ squares . (and there is a root at level 1) 1). This is a tree with every each vertex have has a finite number of childrenschildren. The fact about the positive quadtant quadrant asserts that there are vertices of arbitrary large level, and Konig's theore, theorem then asserts that there is an infinite path which then gives the tiling of the plane. (Proving Konig's theorem is based on moving from a vertex with infinitely many descendants to one of his childs children with infinitely many descandants.descendants.)
I am not sure that this qualifies as an answer since I am not aware of any "finite" proof and there is something infinite in the statement itself. But there may be special cases where a tiling of the positive orthant be explicitely explicitly presented, and it will allow a finite proof which can be very complicated.
