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;
