Can anyone explain this about Directivity relation to effective area?

I am learning Antenna.

http://ptuece.loremate.com/awp/node/2

In question 15, It tries to relate directivity with effective area by using two antenna A1 and A2. I can follow the derivation to the last part where we let

. Then it said according to reciprocity,

. Can anyone explain why this is true even if the two antennas have different directivity and effective area?

Thanks

Alan

It is explained lower in the article.i.e in the reciprocity theorem according to which

"If Antenna A1 is a transmitting antenna and antenna A2 is the receiving one;
the ratio of transmitted to receive Power Pt/Pr will not change if
A1 becomes the receiving antenna and A2 becomes the transmitting one"

Here P1-2 is the receive power when A1 is transmit and A2 is receive antenna and
P2-1 is the receive power when A2 is transmit and A1 is receive antenna

P1 = transmit power by antenna A1
P2 = transmit Power by antenna A2

then according to reciprocity theorem:
P1/P1-2 (Pt/Pr if A1 transmits) = P2/P2-1 (Pt/Pr if A2 transmits)

Now imposing condition P1 = P2;

results in P1-2 = P2-1