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
  |.A - the_midpoint_of_the_segment(A,B).|
    = |. the_midpoint_of_the_segment(A,B) - B.|
  proof
A1: |.A - the_midpoint_of_the_segment(A,B).| = |.A - 1/2*(A+B).| by Th22
                                      .= 1/2*|.A-B.| by Th18;
    |.the_midpoint_of_the_segment(A,B) - B.| = |.1/2*(A+B) - B.| by Th22
                                      .= |.B - 1/2*(A+B).| by EUCLID_6:43
                                      .= 1/2*|.B-A.| by Th18
                                      .= 1/2*|.A-B.| by EUCLID_6:43;
    hence thesis by A1;
  end;
