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.