Einstein

(a) defined a test for clocks at rest wrt each other
    in a stationary system (we'd now say inertial),
    to determine that they are synchronized.  [At
    clock A at time ta send light to clock B which
    reflects it at tb to clock A at ta', with observers
    at each clock noting the time the clock says at the
    three events. If tb-ta=tb-ta' then the clocks are
    synchronized.]

(b) had a stationary system thereby synchronize its
    clocks.

(c) posited a second inertial - but moving - system
    whose clocks at all times and places would show
    the first system's times at the immediately
    adjacent first system location.

(d) posited the first system running the synchroni-
    zation test on the second system clocks; that is, with
    a completely non-definition test.  With r=distance
    between the clocks - per stationary system - he
    got tb-ta=r/(c-v) and tb-ta'=r/(c+v).


He concluded that clocks synchronized in one inertial
system cannot satisfy the definitional test for
synchronization in a second inertial frame.

If the second system had indeed run its synchronization
test like the first system had, the times would be
tb-ta=r/c and tb-ta=r/c.

His proof is much like having a stationary pianist
playing a stationary piano and then turning on his
stationary piano stool to play a second piano that
is moving past him, while he stays stationary.