
theorem Th28:
  for p1,p2,p3,p4 being Point of TOP-REAL 2 st p1`1 <> p3`1 & p4`2
<> p2`2 & p4`2 <=p1`2 & p1`2<=p2`2 & p1`1<=p2`1 & p2`1<=p3`1 & p4`2 <=p3`2 & p3
  `2<=p2`2 & p1`1 <p4`1 & p4`1<=p3`1 holds p1,p2,p3,p4 are_in_this_order_on
  rectangle(p1`1,p3`1,p4`2,p2`2)
proof
  let p1,p2,p3,p4 be Point of TOP-REAL 2;
  set K = rectangle(p1`1,p3`1,p4`2,p2`2);
  assume that
A1: p1`1<>p3`1 and
A2: p4`2 <> p2`2 and
A3: p4`2 <=p1`2 and
A4: p1`2<=p2`2 and
A5: p1`1<=p2`1 and
A6: p2`1<=p3`1 and
A7: p4`2 <=p3`2 and
A8: p3`2<=p2`2 and
A9: p1`1 <p4`1 and
A10: p4`1<=p3`1;
  p4`2 <=p2`2 by A3,A4,XXREAL_0:2;
  then
A11: p4`2 <p2`2 by A2,XXREAL_0:1;
  p1`1<=p3`1 by A5,A6,XXREAL_0:2;
  then p1`1<p3`1 by A1,XXREAL_0:1;
  then
  LE p1,p2,K & LE p2,p3,K & LE p3,p4,K or LE p2,p3,K & LE p3,p4,K & LE p4
  ,p1,K or LE p3,p4,K & LE p4,p1,K & LE p1,p2,K or LE p4,p1,K & LE p1,p2,K & LE
  p2,p3,K by A3,A4,A5,A6,A7,A8,A9,A10,A11,Th4,Th8,Th11;
  hence thesis by JORDAN17:def 1;
end;
