Transformation of two squares in a square

We use the squares in the figure:


Draw segment BE and perpendicular straight lines r and s to the EB passing, respectively, to points I and H.
Let  H be the point of r on segment FG.
Draw the perpendicular straight line to the EH passing through H and let I be the point of intersection of this straight line with s. 
Draw IJ  perpendicular to the BG.

The colors indicate the pieces to transform the two squares given, ABCD and ECGF, into the biggest square  BEHI.
As before, intuitively we can see that the parts cut in the two lesser squares are equal in the bigger square. Formally we must show the congruences between the pairs of triangles LAB and MJI, HGM and EDL, and FEH and JBC.