We are given the equation:
\[ \frac{c}{(a+c)(a-c)} - b(2a - b) - (a - bc)(q - bc) = 0 \]

To prove this, let’s simplify each term step by step.

1. **First Term**: Simplify \(\frac{c}{(a+c)(a-c)}\)
\[
\frac{c}{(a+c)(a-c)} = \frac{c}{a^2 - c^2}
\]

2. **Second Term**: Simplify \(b(2a - b)\)
\[
b(2a - b) = 2ab - b^2
\]

3. **Third Term**: Simplify \((a - bc)(q - bc)\)
\[
(a - bc)(q - bc) = aq - abc - bcq + b^2c^2
\]

Now combine these simplified terms:
\[ \frac{c}{a^2 - c^2} - (2ab - b^2) - (aq - abc - bcq + b^2c^2) = 0 \]

To progress further, we need to reconsider the equation context. There might be constraints or additional relationships between the variables that simplify the equation directly to zero. Let's check if setting some variables to specific values simplifies this expression:

1. **Assign values to simplify**:

For simplicity, assume \(c = a\), let’s substitute and check:
\[
\frac{a}{(a+a)(a-a)} - b(2a - b) - (a - ab)(q - ab) = 0
\]

Note that this leads to an undefined term in \(\frac{a}{0}\).

But in a more straightforward approach:

Consider when \(a = b\) and \(q = a + c\):
\[
\frac{c}{(a+c)(a-c)} - a(2a - a) - (a - a^2)(a + c - a^2) = 0
\]

\[
\frac{c}{a^2 - c^2} - a^2 + 0 = 0
\]

Clearly, plugging \(b = a, q = a+c \), results in simplification issues.

Generally, it seems the solution's intricacies likely only resolved using non-trivial algebraic identities or precise problem constraints homework confirms.

Thus, no standard outright verification without specific clarifying variable relations simplifying to proves all terms correctly neutralizing equals zero.