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;

theorem Th11:
  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
A3:   A in LSeg(B,C);
      dist(B,C) =dist(q,r) & dist(B,A) = dist(q,p) & dist(A,C) = dist(p,r)
          by A2,TOPREAL6:def 1;
      then dist(q,r) = dist(q,p)+dist(p,r) by A3,Th10;
      hence p is_between q,r by A1,METRIC_1:def 22;
    end;
    assume p is_between q,r; then
A4: dist(q,r)=dist(q,p)+dist(p,r) by METRIC_1:def 22;
    dist(q,r)=|.B-C.| & dist(q,p)=|.B-A.| & dist(p,r)=|.A-C.| by A2,SPPOL_1:39;
    hence A in LSeg(B,C) by A4,Th9;
  end;
