# Call option pricing using CCR model - derivation problem

I'm viewing the following derivation of a Call Option price using the CRR model. There is one piece of the derivation which I cannot understand.

\begin{align} C_0 &= e^{-rT} \sum_{i=0}^{N} (S_{0}\,u^{N-i}\,d^{i} - K)^{+} \binom {N}{i} q^{N-i}(1-q)^{i}\\ &= e^{-rN \Delta t} \sum_{i=a}^{N} (S_{0}\,u^{N-i}\,d^{i} - K) \binom {N}{i} q^{N-i}(1-q)^{i}\\ &= S_0 \sum_{i=a}^{N} \binom {N}{i} (u\,q\,e^{-r \Delta t})^{N-i}\, (d\,e^{-r \Delta t}\,(1- q))^{i} - Ke^{-rT} \sum_{i=a}^{N} \binom {N}{i} q^{N-i} (1-q)^{i}\\ &= S_0 \sum_{i=a}^{N} \binom {N}{i} \overline{q}^{N-i}\, (1 - \overline{q})^{i} - Ke^{-rT} \sum_{i=a}^{N} \binom {N}{i} q^{N-i} (1-q)^{i}\\ &= S_0 \mathcal{Q}_1 - K e^{-rT} \mathcal{Q}_2 \end{align}

where $\overline{q} = uqe^{-r\Delta t}$.

$\textbf{Question}$

If $\overline{q} = uqe^{-r\Delta t}$, then I'm assuming $(1-\overline{q}) = d\,e^{-r \Delta t}\,(1- q)$, however I cannot seem to derive this equality.

Appreciate any help understanding why.

Many thanks,

John