Remember, if an aircraft is in level flight then Lift = Weight.

In the scenario given this is the case. So if we just put some figures in as an example.

Just for ease, shall we say that S = 2, Rho = 10, Cl = 1 and V = 100. So to start with we have:

Lift = 1 x 0.5 x 10 x (100 x 100) x 2 = 100000

Now, if we double the speed to 200 and leave everything else the same we have:

Lift = 1 x 0.5 x 10 x (200 x 200) x2 = 400000

But we know that the aircraft is maintaining level flight and therefore Lift cannot change. Hence we need to modify something on the right hand side of the equation so that lift remains at the value of 100000. We can't modify S (only engineers do that) and Rho remains the same as there is nothing to say that the air density has changed. Therefore the only factor we can change is Cl. If we multiply Cl on the right by 0.25 we can ensure that Lift remains at 100000.

You MUST understand that in straight and level flight, lift does not change as it is only balancing against weight (which for this example you can assume to be constant).

So if density remains the same, if IAS doubles TAS doubles too.