By Fubini the plane integral of g(x)h(y) over any rectangle factors as the product of the two coordinate integrals. Additivity of the plane integral over the tiling — valid since distinct tiles meet only in their measure-zero boundaries — writes \int\!\!\int_R g\,h as the sum over tiles of \int\!\!\int_{T_i} g\,h. Each summand is a product with a vanishing factor, so it is 0; hence \bigl(\int_R g\bigr)\bigl(\int_R h\bigr) = 0, and a product of reals (or complex numbers) vanishes only if a factor does.

Take g(x) = h(x) = e^{2\pi i x}. The one-dimensional integral \int_a^b e^{2\pi i x}\,dx = (e^{2\pi i b} - e^{2\pi i a})/(2\pi i) vanishes if and only if e^{2\pi i(b-a)} = 1, i.e. b - a \in \mathbb{Z}. So a tile's coordinate integral vanishes exactly when that side is an integer; the hypothesis gives the per-tile dichotomy, the engine Lemma 7.2 transports it to R, and the same criterion reads off an integer side. The complex exponential is what makes the criterion an exact "integer side" statement, with no reflected solutions.

Use the real integrand \sin(2\pi(x - R_x))\sin(2\pi(y - R_y)), where (R_x, R_y) is the corner of R; shifting the integrand by the corner replaces Wagon's "place R in standard position". The factor integral is \int_a^b \sin(2\pi(x-s))\,dx = (\cos 2\pi(a-s) - \cos 2\pi(b-s))/(2\pi). It vanishes whenever b - a \in \mathbb{Z} (used for the tiles, where the shift is irrelevant); and at the corner, where the lower limit equals the shift, it reduces to (1 - \cos 2\pi(b-s))/(2\pi), which vanishes if and only if b - s \in \mathbb{Z}. The engine Lemma 7.2 then forces an integer side of R.

Colour the plane in a \tfrac12 \times \tfrac12 checkerboard with a corner at R's corner; "equal black and white" over a region means \int\!\!\int (-1)^{\lfloor 2x\rfloor}(-1)^{\lfloor 2y\rfloor} = 0. The one-dimensional factor (-1)^{\lfloor 2(x-s)\rfloor} is the \pm 1 square wave of period 1; its integral over any integer-length interval is 0, so every tile with an integer side is balanced. By the engine Lemma 7.2 so is R. But over [s, b] the square wave integrates to the triangle wave \min(r, 1-r), with r the fractional part of b - s, which is nonzero unless b - s \in \mathbb{Z}; hence R has an integer side.

Disjointness: a point common to two half-open cells moves down-and-left — to the midpoint between the higher of the two lower-left corners and the point itself — into the interiors of both tiles, contradicting interior-disjointness. Covering: a point of the ambient half-open cell is approached from below-left; every such nudge stays in R, hence in some tile, and since there are finitely many tiles one tile contains nudges arbitrarily close in. That tile is closed, so it contains the point, and the nudges witness the two strict inequalities of its half-open cell.

The area of a rectangle is the Lebesgue measure of its half-open cell, and the half-open cells of the tiles partition that of the tiled rectangle Lemma 7.6.

By half-open cells Lemma 7.6. Pull a point z back to the point whose abscissa is the leftmost vertical line of the tiling whose image is not to the left of z, or a point beyond the tiling if there is none, and whose ordinate is the lowest horizontal line whose image is not below z. Since the moves keep the order of the lines, this point lies in the half-open cell of a tile, or of R, exactly when z lies in that of its image. So the half-open cells of the moved tiles are pairwise disjoint and cover the half-open cell of the moved R, because those of the tiling are and do; a tile collapsed to a segment has an empty half-open cell and is discarded.

Floor and ceiling agree at the integers and differ by one everywhere else, so c(x) is 2\lfloor x - a\rfloor in the first case and 2\lfloor x - a\rfloor + 1 in the second.

The translation keeps the order of the lines, so the translated tiles tile the translated rectangle Lemma 7.8. Count the squares by their upper right corners: a square lies in a rectangle whose edges are grid lines exactly when its upper right corner lies in the half-open cell of the rectangle, so every square of the translated rectangle lies in exactly one translated tile Lemma 7.6, and a rectangle whose edges are the grid lines c_0 \le c_1 and d_0 \le d_1 half-steps from the corner of R contains (c_1 - c_0)(d_1 - d_0) squares.

The endpoints of a side of integer length n are n apart, so the translation moves them equally and the increment of the grid coordinate across that side is 2n: every translated tile contains an even number of squares, and hence Lemma 7.10 so does the translated rectangle. The corner of R has grid coordinate 0, so that count is the product of the grid coordinates of the far edges of R, and one of the two factors is even — which says Lemma 7.9 that the corresponding side of R has integer length. (Wagon phrases the last step as a contradiction: a translated rectangle with no integer side has both sides equal to half an odd integer, hence an odd number of squares.)

For t in an interval [0, \delta) the moved lines keep their order, so the moved tiles tile the moved rectangle Lemma 7.8 and the area of the moved rectangle is the sum of theirs Lemma 7.7. The two polynomials therefore agree at infinitely many points, so they are equal.

Let \varepsilon_x(u) be 0 when u - R_x \in \mathbb{Z} and 1 otherwise, and similarly define \varepsilon_y, so that the lines move by \varphi_t(u) = u + t\varepsilon_x(u) and \psi_t(v) = v + t\varepsilon_y(v). The area of a moved rectangle S is the value at t of the polynomial P_S = (\Delta\varepsilon_x X + w)(\Delta\varepsilon_y X + h), with w, h the sides of S and \Delta\varepsilon the indicator increments across them. If a tile has an integer side, the two endpoints of that side have the same indicator, the corresponding factor is constant, and the quadratic coefficient vanishes: the area of the moved tile is linear or constant in t, and hence Lemma 7.12 so is the area of the moved R. If neither side of R is an integer, each lower endpoint has indicator 0 and each upper endpoint 1, so the moved R is (w + t) \times (h + t) and its area has quadratic coefficient 1, a contradiction.

A tile side of integer length n rounds to one of length \lfloor p x_0 + pn\rfloor - \lfloor p x_0\rfloor = pn, so the area of every rounded tile is a multiple of p, and the area of the rounded rectangle is the sum of these areas Lemma 7.8 Lemma 7.7.

The sides of the rounded rectangle are integers that differ from those of the scaled-up rectangle by less than 1. If one of them is 0, the corresponding side of R is shorter than 1/p. Otherwise the area of the rounded rectangle is a multiple of p Lemma 7.14, and p being prime, so is one of its sides, say \lfloor p x_1\rfloor - \lfloor p x_0\rfloor = pm; then p(x_1 - x_0) - pm is a difference of two floor remainders, each in [0,1), so |width - m| < 1/p.

Suppose neither side of R is an integer. Each of the width and the height then keeps some fixed positive distance from every integer — at least the smaller of its fractional part and one minus it. Choosing a prime p larger than the reciprocals of both distances (there are infinitely many primes), no side of R can be within 1/p of an integer, contradicting the claim Lemma 7.15 for this p.

Look at the four quadrants at the point. A tile containing the point fills one of them if the point is a corner of the tile, two if the point lies inside one of its edges, and four if it lies inside the tile. Each quadrant inside R is filled by exactly one tile, so the numbers of quadrants the tiles fill add up to the number of quadrants inside R: four at a point inside R, two at a point inside one of its edges, one at a corner, and none outside R. The tiles having the point as a corner fill one quadrant each and the other tiles an even number, so the former are at most as many as the quadrants inside R, and as many up to an even number.

The vertices that walks from it reach form a union of connected components, so every edge at one of them has both of its ends among them, and their degrees add up to twice the number of these edges. That sum is even, so besides the degree of the starting vertex another of its terms is odd.

The degree of a vertex is the number of tiles having it as a corner, so by the local picture Lemma 7.17 the vertices of odd degree are the corners of R. The lower-left corner is one of them, and by the handshake lemma Lemma 7.18 a walk leads from it to another vertex of odd degree, which is another corner of R.

An edge of \Gamma is a side of a tile, parallel to an axis and of integer length, so it leaves the pair of fractional parts of a point unchanged, and hence so does a walk. The walk to a second corner of R Lemma 7.19 therefore preserves the fractional part of the abscissa or of the ordinate — of both, if it ends at the opposite corner — which says that the width or the height of R is an integer.

Every edge has exactly one end in each part, so either sum counts every edge exactly once.

The two ends of a side of integer length are both on the lattice or neither is, so the corners of the tile on the lattice come in pairs, the two ends of the same side.

Counted from the side of the tiles, the edges of the graph are evenly many Lemma 7.22, so by the count from the other side Lemma 7.21 the numbers of tiles the points of S lie on have an even sum. The lower-left corner of R lies on exactly one tile Lemma 7.17, so it is in S, and the numbers for the other points of S have an odd sum.

Take a point of S other than the lower-left corner of R lying on an odd number of tiles Lemma 7.23. A point that is not a corner of R lies on none, two or four tiles Lemma 7.17, so this one is another corner of R, and being on the lattice through the lower-left corner it is an integer width or height away from it.

Length along the line y = t is a measure on the plane, so it adds up over the half-open cells of the tiles, which partition that of R Lemma 7.6. Because t avoids the horizontal edges, the line meets the half-open cell of a tile, in an interval as long as the tile is wide, exactly when it crosses the tile; and it meets the half-open cell of R in an interval of the full width, since it is not the bottom edge of R: the tile containing the point of the left edge of R at height t reaches below it.

A horizontal line through R avoiding the finitely many heights of horizontal tile edges crosses tiles whose widths add up to the width of R Lemma 7.25, and these widths are integers. For heights, transpose the tiling.

Cut each tile into unit pieces along a side of integer length, of which it has at least one, and which is at least one unit long since the tile has positive width and height. The half-open cells of the pieces of a tile partition its own, so the cells of all the pieces still partition those of R.