
theorem Th45:
  for p1,p2,p3,p4 being Point of TOP-REAL 2, a,b,c,d being Real
 st a<b & c < d & p1`1=b & p2`1=b & p3`1=b & p4`2= c & d >=p1`2 & p1`2>p2
  `2 & p2`2>p3`2 & p3`2>= c & a<p4`1 & p4`1<=b 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`1=b and
A4: p2`1=b and
A5: p3`1=b and
A6: p4`2= c and
A7: d >=p1`2 and
A8: p1`2>p2`2 and
A9: p2`2>p3`2 and
A10: p3`2>= c and
A11: a<p4`1 and
A12: p4`1<=b;
A13: p2`2> c by A9,A10,XXREAL_0:2;
A14: d >p2`2 by A7,A8,XXREAL_0:2;
  then d >p3`2 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,A12,A13,A14,Th10,Th11;
  hence thesis by JORDAN17:def 1;
end;
