theorem Th18:
  |.A - 1/2 *(A+B).| = 1/2 * |.A-B.|
  proof
    |.A-(1/2 * (A+B)).| = |.A -(1/2 * A + 1/2 *B).| by RLVECT_1:def 5
                       .= |.A - 1/2 * A - 1/2 * B.| by RLVECT_1:27
                       .= |. 1 * A - 1/2 * A - 1/2 * B.| by RLVECT_1:def 8
                       .= |.(1-1/2) * A - 1/2 * B.| by RLVECT_1:35
                       .= |.1/2 * (A-B).| by RLVECT_1:34
                       .= |.1/2.| * |.A - B.| by TOPRNS_1:7
                       .= 1/2 * |.A - B.| by ABSVALUE:def 1;
    hence thesis;
  end;
