Let a point object O is placed on the principal axis
By lens formula for image formation by lens L1
v′1−u1=f11 ....... (1)
I' serves as a virtual object for the second lens L2 which forms a final image I at a distance 'v'. Then for L2
v1−v′1=f21 ........ (2)
Adding equation (1) and (2)
v1−u1=f11+f21 .......... (3)
If we replace these two lenses by a single lens such that if an object is kept at u image is formed at v then its focal length f is given by
v1−u1=f1 ....... (4)
From equation (3) and (4) we get
f1=f11+f21