What is the flaw in this proof?

Question

Below is a proof that straight lines cannot exist in the coordinate plane. Where is the flaw in its reasoning?

It will be shown that the equation of a straight line leads to a mathematical inconsistency. First, write the equation of the line in the form Ax+By=C, where A, B, and C are scalars. Second, choose variables u and v so that u=Ax+By and v is any arbitrary non-constant function of x and y. The equation of the line in the uv-plane can be written as u=C. Differentiate implicitly to achieve the equation 1=0, which is a mathematical inconsistency and hence the line cannot exist in the first place.

Answer 1

$u=C$ means $u$ is constant

so after differentiation it should be

$0=0$ not $1=0$.