How to prove that GDP grows at the same rate as consumption on Lab Equipment Model?
The household side of this model is pretty standard. Denote $K(t)$ be household's assets at time $t$. Then the transversality condition (which is an optimizing condition, not a constraint), is 

$$\lim_{t \to \infty} [e^{-\rho t}\lambda(t) K(t)] = 0$$

where $\lambda(t)$ is the current value multiplier on assets in the Hamiltonian. Given the assumed form of the utility function we have the f.o.c.

$$\lambda(t) = \frac {1}{c(t)^{\theta}}$$

So we want 

$$\lim_{t \to \infty} [e^{-\rho t}\frac {1}{c(t)^{\theta}} K(t)] = 0$$

and since in the model we have a constant consumption growth rate, $g_c$ we have

$$c(t)^{\theta} = c(0)^{\theta}\cdot e^{g_c\theta t}$$

Ignoring inconsequential constants, we require

$$\lim_{t \to \infty} [\frac {1}{e^{(\rho+g_c\theta) t}} K(t)] = 0$$

Use the relation between household assets  and the number of goods $N$, the relation between income (from labor and assets) to aggregate output, as well as the expression of aggregate output as a function of $N$, to obtain a first-order linear differential equation in $N$, and proceed  as in pages 208-209 of the book.