The generating functional for the connected part of the Green functions is defined as
$$iW[j] = \log Z[j].$$
From this the four-point connected Green's function is then given by $G_c(x_1,x_2,x_3,x_4)^{(4)}=G^{(4)}(x_1,x_2,x_3,x_4)-G_c^{(2)}(x_1,x_2)G_c^{(2)}(x_3,x_4)-G_c^{(2)}(x_1,x_3)G_c^{(2)}(x_2,x_4)-G_c^{(2)}(x_1,x_4)G_c^{(2)}(x_2,x_3) \tag 1$
where superscript means functional derivative $W[j]$ in respect to $j$ that is
$$G_c^{(i)}=\frac{\delta }{\delta j_1...\delta j_1}W[j]$$
and
$$G^{(i)}=\frac{\delta }{\delta j_1...\delta j_1}Z[j].$$
From equation $(1)$ we can see that the connected part of $G^{(4)}(x_1,x_2,x_3,x_4)$ is contained in $G_c(x_1,x_2,x_3,x_4)^{(4)}$ but how can we prove that $G_c(x_1,x_2,x_3,x_4)^{(4)}$ does contain only connected parts?
