See the following figure:

First obtain the output functions for the S-R latch (top figure).
Break the feedback loop at Q1; the output functions are thus

Q1+ = (S Q2)' = S' + Q2' = S' + Q1 R Q2 = (Q1 R)' = Q1' + R'

or you can examine the truth table:

S R | Q1+ Q2 -------+--------- 0 0 | 1 1 0 1 | 1 0 1 0 | 0 1 1 1 | Q1 Q1'

Note that we must be careful about the output Q2; it isn't necessarily the complement of Q1 !

Now look at the bottom figure; this is the given circuit with the
S-R latches replacing the cross-coupled NANDs. Substituting using the
above characteristic equations for the S-R latches, we get:

Y1+ = S1' + R1 Y1 = X1 + Y4 Y1 = X1 + (Y2' + R2') Y1 = X1 + (Y2' + X2) Y1 = X1 + Y2' Y1 + X2 Y1 Y2+ = S2' + R2 Y2 = (X1' Y3)' + X2' Y2 = X1 + Y3' + X2' Y2 = X1 + (Y1' + R1')' + X2' Y2 = X1 + Y1 Y4 + X2' Y2 = X1 + Y1 (Y2' + R2') + X2' Y2 = X1 + Y1 Y2' + Y1 X2 + X2' Y2

The flow table looks like: (asterisks show stable positions)

(Y1+ Y2+) (X1 X2) 00 01 11 10 (Y1 Y2) 00 00* 00* 11 11 01 01* 00 11 11 11 01 11* 11* 11* 10 11 11 11 11

Note that there is a race condition (as (Y1 Y2) goes from (00) to (11)), but it is not critical.