 reserve n for Nat;

theorem ThAZ9:
  for P,Q,R being Point of TOP-REAL 2, L being Element of line_of_REAL 2
  st P in L & Q in L & R in L holds P in LSeg(Q,R) or
    Q in LSeg(R,P) or R in LSeg(P,Q)
  proof
    let P,Q,R be Point of TOP-REAL 2, L be Element of line_of_REAL 2;
    assume that
A1: P in L and
A2: Q in L and
A3: R in L;
    L in line_of_REAL 2;
    then L in the set of all Line(x1,x2) where x1,x2 is Element of REAL 2
      by EUCLIDLP:def 4;
    then consider x1,x2 be Element of REAL 2 such that
A4: L = Line(x1,x2);
    reconsider tx1 = x1,tx2 = x2 as Element of TOP-REAL 2 by EUCLID:22;
    P in Line(tx1,tx2) & Q in Line(tx1,tx2) & R in Line(tx1,tx2)
      by A1,A2,A3,A4,EUCLID12:4;
    hence thesis by RLTOPSP1:def 16,TOPREAL9:67;
  end;
