FI-MO 2009 Problem 5

As in the picture below, the rectangle on the left hand side has been divided into four parts by line segments which are parallel to a side of the rectangle. The areas of the small rectangles are $A,B,C$ and $D$ . Similarly, the small rectangles on the right hand side have areas $A^\prime,B^\prime,C^\prime$ and $D^\prime$ . It is known that $A\leq A^\prime$ , $B\leq B^\prime$ , $C\leq C^\prime$ but $D\leq B^\prime$ .

Prove that the big rectangle on the left hand side has area smaller or equal to the area of the big rectangle on the right hand side, i.e. $A+B+C+D\leq A^\prime+B^\prime+C^\prime+D^\prime$ .