Circles S1 and S2 intersect at points M and N. Through point A of circle S1, draw straight lines AM and AN intersecting S2 at points B and C, and through point D of circle S2, draw straight lines DM and DN intersecting S1 at points E and F, and A, E, F lie along one side of line MN, and D, B, C lie on the other side (see figure). Prove that if AB=DE, then points A, F, C and D lie on the same circle, the position of the center of which does not depend on choosing points A and D.