
theorem
  for p1, p2, q1, q2, q3 being Point of TOP-REAL 2 st p1 <> p2 &
  LE q1,q2,p1,p2 & LE q2,q3,p1,p2 holds LE q1,q3,p1,p2
proof
  let p1, p2, q1, q2, q3 be Point of TOP-REAL 2;
  assume that
A1: p1<>p2 and
A2: LE q1,q2,p1,p2 and
A3: LE q2,q3,p1,p2;
A4: q1 in LSeg(p1,p2) by A2;
A5: q2 in LSeg(p1,p2) by A2;
A6: q3 in LSeg(p1,p2) by A3;
  consider s1 being Real such that
A7: q1=(1-s1)*p1+s1*p2 and 0<=s1 and
A8: s1<=1 by A4;
  consider s2 being Real such that
A9: q2=(1-s2)*p1+s2*p2 and
A10: 0<=s2 and
A11: s2<=1 by A5;
A12: s1 <= s2 by A2,A7,A8,A9,A10,A11;
  consider s3 being Real such that
A13: q3=(1-s3)*p1+s3*p2 and
A14: 0<=s3 and
A15: s3<=1 by A6;
  s2 <= s3 by A3,A9,A11,A13,A14,A15;
  then
A16: s1 <= s3 by A12,XXREAL_0:2;
  thus LE q1,q3,p1,p2
  proof
    thus q1 in LSeg(p1,p2) & q3 in LSeg(p1,p2) by A2,A3;
    let r1,r2 be Real;
    assume that
    0<=r1 and r1<=1 and
A17: q1=(1-r1)*p1+r1*p2 and 0<=r2
    and r2<=1 and
A18: q3=(1-r2)*p1+r2*p2;
    s1 = r1 by A1,A7,A17,JORDAN5A:2;
    hence thesis by A1,A13,A16,A18,JORDAN5A:2;
  end;
end;
