k_{\perp}^2)
A_j^{(QED)}(s\beta/ k_{\perp}^2) ~,$$

$$\label{qqedtilde}
 \tilde{Q}_j^{(QED)}(s/ M^2) = 
c_j \int_{M^2}^s 
d k_{\perp}^2 \int \frac{d \alpha}{\alpha}\frac{ d \beta}{\beta} 
\frac{k_{\perp}^2 }{(s\alpha\beta - k_{\perp}^2)^2} 
 \tilde{A}_j^{(QED)}(s\alpha/ k_{\perp}^2)
 \tilde{A}_j^{(QED)}(s\beta/ k_{\perp}^2) ~,$$

$$\label{qtilde}
 \tilde{Q}_j(s/ M^2) = 
c_j \int_{M^2}^s 
d k_{\perp}^2 \int \frac{d \alpha}{\alpha}\frac{ d \beta}{\beta} 
\frac{k_{\perp}^2 }{(s\alpha\beta - k_{\perp}^2)^2} 
 \tilde{A}_j(s\alpha/ k_{\perp}^2)
 \tilde{A}_j(s\beta/ k_{\perp}^2) ~,$$

and $$\begin{aligned}
\label{qprime} 
 {Q'}_j(s/M^2, \eta) = 
c_j \int_{\mu^2}^{M^2} 
d k_{\perp}^2 \int \frac{d \alpha}{\alpha}\frac{ d \beta}{\beta} 
\frac{2k_{\perp}^2 }{(s\alpha\beta - k_{\perp}^2)^2} 
\Big(2 A_j^{(QED)}(s\alpha/ k_{\perp}^2 )
 {A'}_j(s\beta/ k_{\perp}^2, M^2/k_{\perp}^2) \\ \nonumber 
 + {A'}_j(s\alpha/ k_{\perp}^2, M^2/k_{\perp}^2) 
 {A'}_j(s\beta/ k_{\perp}^2, M^2/k_{\perp}^2) \Big)~. \end{aligned}$$

Now we are able to write the IREE for amplitudes $A_j$. The general form it given by: $$\label{ireegeneral}
A_j = B_j + G_j + Q_j ~.$$

Then using Eqs. (\[gtot\]) and (\[qtot\]) we can rewrite it as $$\label{iree}
{A'}_j + A_j^{(QED)}= B_j^{(QED)} - \tilde{B}_j^{(QED)} + \tilde{B}_j 
+ G_j^{(QED)} - \tilde{G}_j^{(QED)} + \tilde{G}_j + {G'}_j
+ Q_j^{(QED)} - \tilde{Q}_j^{(QED)} + \tilde{Q}_j + {Q'}_j ~.$$

Let us notice that $A_j^{(QED)}(s/\mu^2)$ obeys the equation $$\label{ireeqed}
A_j^{(QED)} = B_j^{(QED)} + G_j^{(QED)} + Q_j^{(QED)}$$ and therefore $A_j^{(QED)}$ cancels out in Eq. (\[iree\]). Also, the auxiliary amplitudes $\tilde{A}_j$ and $\tilde{A}_j^{(QED)}$, obey similar equations: $$\label{ireem}
\tilde{A}_j^{(QED)} = \tilde{B}_j^{(QED)} + \tilde{G}_j^{(QED)} + 
\tilde{G}_j^{(QED)}, ~~
\tilde{A}_j = \tilde{B}_j + \tilde{G}_j + \tilde{Q}_j ~.$$

The 