 reserve n for Nat;

theorem THNOIX:
  for A,B,C,D being Point of TOP-REAL 2 st
    B in LSeg(A,C) & D in LSeg(A,B) holds
      B in LSeg(D,C)
  proof
    let A,B,C,D be Point of TOP-REAL 2;
    assume that
A1: B in LSeg(A,C) and
A2: D in LSeg(A,B);
A3: dist(A,D) + dist(D,C) = dist(A,C) by A1,A2,THORANGE,EUCLID12:12;
A4: dist(A,B) + dist(B,C) = dist(A,C) by A1,EUCLID12:12;
    dist(A,D) + dist(D,B) = dist(A,B) by A2,EUCLID12:12;
    then dist(D,B) + dist(B,C)= dist(D,C) by A3,A4;
    hence thesis by EUCLID12:12;
  end;
