theorem
  for p,q,r be Element of Euclid 2 st
  p,q,r are_mutually_distinct & p=A & q=B & r=C holds
  A in LSeg(B,C) iff p is_Between q,r
  proof
    let p,q,r be Element of Euclid 2;
    assume that
A1: p,q,r are_mutually_distinct and
A2: p = A & q = B & r = C;
    hereby
      assume A in LSeg(B,C);
      then p is_between q,r by A1,A2,Th11;
      hence p is_Between q,r by METRIC_1:def 22;
    end;
    assume p is_Between q,r;
    then p is_between q,r by A1,METRIC_1:def 22;
    hence A in LSeg(B,C) by A1,A2,Th11;
  end;
