will be clear from context which incidence is which. Following a standard convention we call the points and lines of the complement $\fixoutf(\A,\W)$ *proper*, and those points and lines of $\A$ that are not in $\pointsout$, $\linesout$ respectively are said to be *improper*. The set $\W$ will be called the *horizon of $\fixoutf(\A,\W)$*. Among proper lines of $\A$ we distinguish those which intersect the horizon $\W$ and call them *affine*. A triangle in $\A$ is said to be *proper* if all its vertices and sides are proper.

Let us introduce, quite critical for our study, the notion of *the index of $\W$ in $\A$*. It will be denoted by $\indf(\A,\W)$ and

$\indf(\A,\W)=\lambda$ iff there exist a line with $\lambda$ improper points\
and every line either has at most $\lambda$ improper points or it is contained in $\W$.

For $\indf(\A,\W) = 0$ we get $\fixoutf(\A,\W)=\A$. We assume in the sequel that $$\label{eq:non-zero-ind}
  0 < \indf(\A,\W) < \infty$$ and $\indf(\A,\W)$ is a well defined integer. There is one more caveat yet. The complement $\fixoutf(\A,\W)$ is not a partial linear space in general as there could be one-element sets in $\linesout$. This defect is ruled out by the assumption that $$\label{eq:size-of-line}
  \text{\itshape every 