Reflection and transmission of EM waves

I'm reading "Introduction to Electrodynamics" by David J. Girffiths and the following assumption is confusing me:

We have an EM wave inciding on a surface,

Then, when applying the boundary conditions, the following is done:

What confuses me, regarding the highlighted part, is, why should those exponentials be equal? Isn't it sufficient for them to be constant at $$z = 0$$?

• If they are equal for all the times, is it not a constant when z = 0? Jan 25 '19 at 19:06
• What have you tried? Suppose $\alpha e^{iax} + \beta e^{ibx} = \gamma e^{icx}$ for all $x$ and for nonzero $\alpha, \beta, \gamma, a, b, c$. What can we say about $a, b, c$? Jan 25 '19 at 19:14

I see that "$$\exp{(ikx)}$$" gave the idea in commentary while I was typing. Simply to complete (with a poor english) :
With $$z = 0$$ and taking $$y = 0$$ you have a linear combination $$A{{e}^{i{{k}_{1x}}x}}+B{{e}^{i{{k}_{2x}}x}}=C{{e}^{i{{k}_{3x}}x}}$$ equality valid for all $$x$$, with $$A$$, $$B$$ and $$C$$ different from $$0$$.
If you agree that $$\left\{ {{e}^{i\alpha x}} \right\}_{\alpha}$$ form a linearly independent set of functions, the only solution is that the argument of the exponential must be equal.