theorem Th22:
  the_midpoint_of_the_segment(A,B) = 1/2 * (A+B)
  proof
    now
      1/2 * (A+B) = (1-1/2) * A + 1/2 * B by RLVECT_1:def 5;
      then 1/2 * (A+B) in {(1-r)*A+r*B:0<=r<=1};
      hence 1/2 * (A+B) in LSeg(A,B) by RLTOPSP1:def 2;
      thus |.A-(1/2 *(A+B)).| = half_length(A,B) by Th18;
    end;
    hence thesis by Def1;
  end;
