reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve x,y,z,y1,y2 for Element of REAL 2;
reserve L,L1,L2,L3,L4 for Element of line_of_REAL 2;
reserve D,E,F for Point of TOP-REAL 2;
reserve b,c,d,r,s for Real;

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;
