Before revision:  We use these three properties to define a pseudospherical drawing of G. Our main result is that, for the complete graph, these three properties are equivalent to the same three properties but with " exactly two crossings"  replaced by "at most two crossings".
After revision:  We use these three properties to define a pseudospherical drawing of G. Our main result is that, for the complete graph, these three properties are equivalent to the same three properties but with " precisely two crossings" replaced by "at most two crossings".