
theorem Th14:
  for p1,p2,p3,p4 being Point of TOP-REAL 2, a,b,c,d being Real
 st a<b & p1`1=a & p2`1=a & p3`1=a & p4`1=a & c <=p1`2 & p1`2<p2`2 & p2`2
<p3`2 & p3`2<p4`2 & p4`2<=d 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: p1`1=a and
A3: p2`1=a and
A4: p3`1=a and
A5: p4`1=a and
A6: c <=p1`2 and
A7: p1`2<p2`2 and
A8: p2`2<p3`2 and
A9: p3`2<p4`2 and
A10: p4`2<=d;
A11: p3`2<d by A9,A10,XXREAL_0:2;
  p2`2<p4`2 by A8,A9,XXREAL_0:2;
  then
A12: p2`2<d by A10,XXREAL_0:2;
A13: c <p2`2 by A6,A7,XXREAL_0:2;
  then c <p3`2 by A8,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,A12,A13,A11,Th3;
  hence thesis by JORDAN17:def 1;
end;
