Hypothesis Testing of the Difference between Proportions

My textbook says:
"The Null Hypothesis will be pi_1=pi_2 (=pi, say) i.e. that the two samples are from the same population. This being so the best estimate of the standard error of the difference of pi_1 and pi_2 is given by pooling the samples and finding the pooled sample proportion thus
pi=(p_1*n_1+p_2*n_2)/(n_1+n_2) and the standard error is:
s_(p_1-p_2)=sqrt(p*q/n_1+p*q/n_2)
and
z=((p_1-p_2)-(n_1-n_2))/s_(p_1-p_2)
but where the Null Hypothesis is pi_1=pi_2 the second part of the numerator disappears."
My question: Where did this -(n_1-n_2) come from?
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I feel comfortable with the logic behind the calculation of the z score in hypothesis testing of the difference between two means:
z=(mean_A-mean_B)/s_(mean_A-mean_B), where
s_(mean_A-mean_B)=sqrt((s_A)^2/n_A+(s_B)^2/n_B), where
s_A - standard deviation of sample A, size n_A and
s_B - standard deviation of sample B, size n_B
Following the same logic I came to the conclusion that the z score in hypothesis testing of the difference between proportions should be
z=(p_1-p_2)/s_(p_1-p_2) and not
z=((p_1-p_2)-(n_1-n_2))/s_(p_1-p_2)

tomma said:

Following the same logic I came to the conclusion that the z score in hypothesis testing of the difference between proportions should be
z=(p_1-p_2)/s_(p_1-p_2) and not
z=((p_1-p_2)-(n_1-n_2))/s_(p_1-p_2)

Click to expand...

I think you are right, at least you have a valid asymtotic (valid for large n's) test.

Thank you