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Proof. Let's notice at first that the statement is not obvious as there can be various systems of the weighed points, such that but it easily follows from that fact that for any couple the ratio is executed

b) If, attracts with a trivial image (as can accept only two values 0. If, - two vectors from, the point determined by a condition is, our statement from where follows.

. In order that the family of points from was affine free (. affinely generating), it is necessary and enough that the family was free (. family forming) in vector space

The theorem Let - the affine space associated with vector space Affine (. bijection poluaffinna on form group which we designate (.). Display (linear or semi-linear part) is homomorphism on and on group of semi-linear bijections on.

The offer Let, - two vector spaces over the same body and (respectively) – affine hyperplane in (.), not passing through the beginning; let's designate (respectively) the vector hyperplane parallel (respectively).

If is LAMAS with the directing subspace and - a point, allows structure of vector space from the beginning and there is a vector subspace in ℰA. Back, any the runway of space of ℰA is LAMAS, passing through; let's formulate

Below we designate through, two affine spaces associated according to vector spaces over any bodies. We will give purely geometrical characterization of semi-affine displays of century. For clarity we will begin with a case of injective displays.