
theorem Th48:
  for p1,p2,p3,p4 being Point of TOP-REAL 2, a,b,c,d being Real
 st a<b & c < d & p1`2= c & p2`2= c & p3`2= c & p4`2= c & b >=p1`1 & p1`1
>p2`1 & p2`1>p3`1 & p3`1>p4`1 & p4`1> a holds p1,p2,p3,p4 are_in_this_order_on
  rectangle(a,b,c,d)
proof
  let p1,p2,p3,p4 be Point of TOP-REAL 2, a,b,c,d being Real;
  set K=rectangle(a,b,c,d);
  assume that
A1: a<b and
A2: c < d and
A3: p1`2= c and
A4: p2`2= c and
A5: p3`2= c and
A6: p4`2= c and
A7: b >=p1`1 and
A8: p1`1>p2`1 and
A9: p2`1>p3`1 and
A10: p3`1>p4`1 and
A11: p4`1> a;
A12: p3`1>a by A10,A11,XXREAL_0:2;
  p2`1>p4`1 by A9,A10,XXREAL_0:2;
  then
A13: p2`1>a by A11,XXREAL_0:2;
A14: b>p2`1 by A7,A8,XXREAL_0:2;
  then b >p3`1 by A9,XXREAL_0:2;
  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 A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A13,A14,A12,Th12;
  hence thesis by JORDAN17:def 1;
end;
