Regarding matsui's Paper on Linear Cryptanalysis of DES I am going through the the Paper by Matsui on Linear Cryptanalysis of DES. In that, he says

$NS_{5}(16,15)=12$

And then in the next paragraph he says considering the expansion and permutation phases the following equation holds good:

$X(15)\bigoplus F(X,K)[7,18,24,29]=k[22]$

Can somebody help me understand this, because I can't find a relation between the first equation and the second one?
Let us denote by $x = x_5x_4x_3x_2x_1x_0$, where $x_i \in \{0, 1\}$ the input of a DES S-box and by $y = y_3y_2y_1y_0$, with $y_i \in \{0, 1\}$ its output. Basically, $\mathrm{NS}_5(16, 15) = 12$ means that for S-box #5, the relation $x_4 = y_3 \oplus y_2 \oplus y_1 \oplus y_0$, where $y = S_5(x)$, holds with probability $\frac{\mathrm{NS}_5(16, 5)}{64} = \frac{12}{64} = \frac{1}{2}-\frac{5}{16}$, which is a fairly biased value (one would expect this value to be very close to $\frac{1}{2}$ in an ideal situation). 

Starting from this observation, Matsui builds the 1-round linear approximation $X_{15} \oplus F(X, K)_7 \oplus F(X, K)_{18} \oplus F(X, K)_{24} \oplus F(X, K)_{29} = K_{22}$ that holds with the same probability $\frac{1}{2}-\frac{5}{16}$. Note that, for S-box #5, $x_4$ can be traced back to $X_{15} \oplus K_{22}$, taking into account the $\mathrm{EP}(.)$ transformation as well as the key-schedule algorithm, and $y_3y_2y_1y_0$ can be propagated to bits 7, 18, 24 and 29 of the output of the round function, this time taking into account the effect of the bit permutation $\mathrm{P}(.)$.