= Solution
The transmitting <antenna> is matched to its input <transmission line>, so its real terminal <electrical resistance> is $Z_i$. The identical receiving <antenna> has the same <electrical impedance> at the operating <frequency>. Its available received <power> for a matched load is the incident <energy flux> times its <antenna effective area>:
$$
P_{\rm av}=\frac{P_iG_a}{4\pi d^2}A_e
=P_i\left(\frac{G_a\lambda}{4\pi d}\right)^2\equiv P_i\mathcal F.
$$
The real receiving load is $Z_t$, so its voltage-wave <reflection coefficient> relative to $Z_i$ is $\Gamma_L=(Z_t-Z_i)/(Z_t+Z_i)$. The fraction of available <power> delivered to the output line is
$$
1-|\Gamma_L|^2=\frac{4Z_iZ_t}{(Z_i+Z_t)^2}.
$$
Consequently, neglecting path losses, mismatch of <polarization of an electromagnetic wave> and line losses,
$$
\boxed{P_t=P_i\left(\frac{G_a\lambda}{4\pi d}\right)^2
\frac{4Z_iZ_t}{(Z_i+Z_t)^2}.}
$$
This is the <Friis transmission equation> with a receiving impedance-mismatch factor. For fixed $P_i,G_a,d,Z_i,Z_t$, it scales as $\lambda^2$. That scaling is conditional on holding <antenna gain> fixed: a fixed physical aperture can instead have $G_a\propto\lambda^{-2}$, and any <wavelength> dependence of the impedances also enters the expression. The matched-output check $Z_t=Z_i$ gives $P_t=P_i\mathcal F$.
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