Surface-Catalyzed Reaction

(Please ignore the strikethroughs, I don't know why they're there...)

The following is a basic surface-catalyzed reaction:
A -> P
It can be modelled by the following mechanism:
A + S <-> A-S
A-S -> P + S

We are trying to derive the rate law for the formation of P, which is
d[P]/dt = k[A-S].
This can also be written as
d[P]/dt = k’ thetaA, where thetaA is the fraction of sites to which a reactant is bound.

The expression for thetaA is
thetaA = [A-S] / ([ S] + [A-S ]).

Since this expression has intermediates, we need to get rid of [S ]and [A-S], by applying steady state approx. to both.
d[A-S ]/dt = k1[A] - (k2 + k-1)[A-S ]
d/dt = - k1[A] + (k2 + k-1)[A-S ]

I don’t understand the following:

Because we have a catalytic cycle, it turns out that the two halves of the cycle fit together perfectly, and our two SSA equations are the same.

Can anyone explain why this is the case? Thank you!