
theorem Th17:
  for q1, q2, p1, p2 being Point of TOP-REAL 2 st p1 <> p2 holds
  LE q1, q2, LSeg(p1, p2), p1, p2 implies LE q1, q2, p1, p2
proof
  let q1, q2, p1, p2 be Point of TOP-REAL 2;
  set P = LSeg (p1, p2);
  assume p1 <> p2;
  then consider f be Function of I[01], (TOP-REAL 2) | LSeg(p1,p2) such that
A1: for x being Real st x in [.0,1.] holds f.x = (1-x)*p1 + x*p2 and
A2: f is being_homeomorphism & f.0 = p1 & f.1 = p2 by JORDAN5A:3;
  assume
A3: LE q1, q2, P, p1, p2;
  hence q1 in P & q2 in P;
  let r1,r2 be Real;
  assume that
A4: 0<=r1 and
A5: r1<=1 and
A6: q1=(1-r1)*p1+r1*p2 and
A7: 0<=r2 & r2<=1 and
A8: q2=(1-r2)*p1+r2*p2;
  r2 in [.0,1.] by A7,BORSUK_1:40,43;
  then
A9: q2 = f.r2 by A8,A1;
  r1 in [.0,1.] by A4,A5,BORSUK_1:40,43;
  then q1 = f.r1 by A6,A1;
  hence thesis by A3,A5,A7,A2,A9;
end;
