theorem Th28:
  C in LSeg(A,B) & |.A-C.| = |.B-C.| implies
  C = the_midpoint_of_the_segment(A,B)
  proof
    assume that
A1: C in LSeg(A,B) and
A2: |.A-C.| = |.B-C.|;
    |.A-C.| = half_length(A,B) by A1,A2,Th27;
    hence thesis by A1,Def1;
  end;
