The proof is kind of long and boring. Here goes:

Consider to sets ox axes: Oxyz, that appears stationary to anyone/anything on earth's surface and O'x'y'z', the in which earth itself is stationary and not rotating. Let O=O'. O'x'y'z' rotation is given by the constant vector w.

a position in O'x'y'z' is r' = x'e(x')+y'e(y')+z'e(z') with e(x') the x' unity vector

therefore, by derivation:

v' = v(x)'e(x')+v(y)'e(y')+v(z)'e(z') with v(x') the projection of v on the x'-vector

and: a' = a(x)'e(x')+a(y)'e(y')+a(z)'e(z')

d(e(z'))/dt is the speed of e(z')'s tip, so = v* = w^R* = w^R = w^e(z')

apply same logic to e(x') and e(y')

the speed of an object with mass m in Oxyz is dr/dt = dr'/dt transformed from O'x'y'z' to Oxyz

dr'/dt = dx'/dt*e(x') + dy'/dt*e(y') + dz'/dt*e(z') + x'dx'/dt + y'dy'/dt + z'dz'/dt

=v'+x'(w^e*x') + y'(w^e*y') + y'(w^e*y')

=v' + w^r'

By derivation:

a=dv/dt =dv'/dt + w^(dr'/dt) both transformed from O'x'y'z' to Oxyz

we still have to look for dv'/dt, using the same strategy as above:

dv'/dt = dv(x')/dt*e(x') + dv(y')/dt*e(y') + dv(z')/dt*e(z') + v(x')(w^e*x') + v(y')(w^e*y') + v(z')(w^e*z')

=a' + (w^v')

And there we have it:

a= a' + )w^v') + w^(v'+ w^r)

= a' + 2w^v' + w^(w^r)

so a' = a - 2w^v' - w^(w^r)

This first term is the actual acceleration of the object on earth's surface, the third term is the good old centrifugal force, the second term is what became known as the Coriolis force. Since I see that Thork has proposed a plausible hypothesis for the storm's rotation on a flat earth, I propose to use Coriolis for the following:

Foucault's pendulum is also explained by this force. However, you can see that on a flat earth, Foucault's pendulum would rotate in the same direction on both hemispheres and even would rotate even at the equator, while on a round earth, it wouldn't rotate at the equator and would rotate in the opposite direction on the northern and southern hemispheres. This is a very easy experiment that will prove the shape of the earth once and for all! All we need is 2 people at different latitudes willing to participate. If both notice exactly the same amount of rotation in the same amount of time, earth is flat. If they don't, earth is round.