I am referring to this document on Backlund transformations.
In this paper, one has equations (50), $$ \begin{align*} (w_1+w_0)_x&=2\lambda_1+\frac{1}{2}(w_1-w_0)^2\\ (w_2+w_0)_x&=2\lambda_2+\frac{1}{2}(w_2-w_0)^2 \end{align*}, $$ and equations (51), $$ \begin{align*} (w_{12}+w_1)_x&=2\lambda_1+\frac{1}{2}(w_{12}-w_1)^2\\ (w_{21}+w_2)_x&=2\lambda_2+\frac{1}{2}(w_{21}-w_2)^2 \end{align*}, $$
Then it is said:
subtract the difference of eqns (50) from the difference of eqns (51) to give $$ 0=4(\lambda_2-\lambda_1)+\frac{1}{2}[(w_{12}-w_1)^2-(w_{21}-w_2)^2-(w_1-w_0)^2+(w_2-w_0)^2] $$
I do not understand what is meant. What do I have to subtract from what? I am confused.
$A=B$
$C=D$
$E=F$
$G=H$
Subtract the difference of first two equations from the difference of second two equations means
subtract $A-C=B-D$ from $E-G=F-H$. So you get $(E-G)-(A-C)=(F-H)-(B-D)$.