The center of mass is the integral of f×v over m, where v is the vector from the origin to the points of s. This is really n integrals combined into one. In 3 dimensions, integrate xf, yf, and zf, divide through by m, and find a point in 3 space. This is the center of mass. We need to show that the coordinates don't matter.
If c is added to x, the integral increases by cm. Divide by m, and c is added to the x coordinate of the center of mass. When s is translated in space, the center of mass moves with it. The origin can be moved to any location you like.
The coordinate basis is usually used to compute the center of mass, but the definition is not linked to that basis. Consider any net over s and compute the Riemann sum. Each point in the net becomes a vector, and the Riemann sum is a linear combination of these vectors, with coefficients determined by f. Any linear combination of vectors, including this linear combination of vectors, gives the same point in space, even if another basis is used to represent those vectors. All Riemann sums are basis invariant, hence the integral, and the center of mass, is basis invariant. We may choose any coordinate system we like.
For a complete, rigorous proof, we need to know that the rectangles in a Riemann net can be tilted, as we change coordinate systems, and that doesn't change the integral. This is discussed elsewhere.
Draw any plane through the center of mass of s. Let x be the perpendicular coordinate. Now the integral of xf must be 0, since the plane passing through x=0 contains the center of mass. In other words, the integral of xf for x positive is the same as the integral of -xf for x negative. On each side of the plane, the integral of xf represents the torque, the weight of all the points of s pulling down on levers of length x. The torque on one side is the same as the torque on the other. If s is supported at its center of mass, in a gravitational field, it will not tilt up or down relative to the x axis. This holds true for all planes passing through the center of mass, hence s is perfectly balanced at its center of mass.