Really, let, - the straight lines in having the same image, let - two various points of their general image. Then prototypes of points also belong and at the same time and are various (owing to an inyektivnost), from where follows that.
For example, there is a bijection of vector space over in vector space over, and the image of each straight line from at display contains in a fnekotory straight line of space, but is not semi-linear (as are not isomorphic).
Proof. On a lemma 2, and an essence of LAMAS of century. Assuming that, we fix a point in and a point in; we will designate parallel translation on a vector through. For any point the straight line is parallel to a straight line and as the image of a straight line is reduced to one point, the image of a straight line is reduced to one point. Thus, attracts and inclusion takes place.
Remark. Conditions of the theorem 1 are satisfied, in particular, if injective display in themselves, such that the image of any straight line is direct, parallel; then it is possible to prove directly that dilatation.
From this it follows that meets conditions and, imposed on, on condition of replacement on. The lemma 4 shows then that images at display of two parallel straight lines, from - two parallel straight lines. At last, meets all conditions of the theorem 1 (after replacement on). Therefore, poluaffinno and also business of page is.
This result is especially interesting in a case when bodies both coincide and do not allow other automorphisms, except identical (for example, when or at: in this case we receive purely geometrical characterization of affine displays of a rank of space of century.
Proof. The beginning choice in reduces business to a case of a faktorprostranstvo of vector space On its vector subspace, and it appears that it is enough to apply the theorem 3, having accepted a point for the beginning of century.
For example, there is a bijection of vector space over in vector space over, and the image of each straight line from at display contains in some straight line of space, but is not semi-linear (as are not isomorphic).
Proof. The result is obvious if it is reduced to one point. Otherwise for any couple of various points, the straight line contains in agrees. Thus, the straight line contains in and the theorem 8 shows that is LAMAS.