reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;

theorem
  for p1,p2,q1 being Point of TOP-REAL n st (p1 <> p2 or p2 <> q1) &
LSeg(p1,p2) /\ LSeg(p2,q1) = {p2} holds LSeg(p1,p2) \/ LSeg(p2,q1) is_an_arc_of
  p1,q1
proof
  let p1,p2,q1 be Point of TOP-REAL n;
  assume that
A1: p1 <> p2 or p2 <> q1 and
A2: LSeg(p1,p2) /\ LSeg(p2,q1) = {p2};
  per cases by A1;
  suppose
    p1 <> p2;
    hence thesis by A2,Th9,Th10;
  end;
  suppose
    p2 <> q1;
    hence thesis by A2,Th9,Th11;
  end;
end;
