Proving that a shape S is a sphere.

Let $\displaystyle S\subset {\mathbb{R}}^3$ with the following properties:

1. For any line $\displaystyle l$:

$\displaystyle |l\cap S|=2$ or $\displaystyle |l\cap S|=1$ or $\displaystyle |l\cap S|=0$

2. For any plane$\displaystyle P$:

$\displaystyle |P\cap S|=4$ or $\displaystyle |P\cap S|=1$ or $\displaystyle |P\cap S|=0$

Prove that $\displaystyle S$ is a sphere.

Re: Proving that a shape S is a sphere.

What does the | | sign mean? First I thought it must be the number of elements of the set. But in that case |PS| = infinity, not 4.