
theorem Th6:
  for p1,p2 being Point of TOP-REAL 2, b,d being Real st p2
`2 <d & p2`2 <=p1`2 & p1`2<=d & p1`1<p2`1 & p2`1<=b holds LE p1,p2,rectangle(p1
  `1,b,p2`2,d)
proof
  let p1,p2 be Point of TOP-REAL 2, b,d be Real;
  set a=p1`1, c =p2`2, K=rectangle(a,b,c,d);
  assume that
A1: c <d and
A2: c <=p1`2 and
A3: p1`2<= d and
A4: a<p2`1 and
A5: p2`1<=b;
A6: p1 in LSeg(|[a,c]|,|[a,d]|) by A1,A2,A3,JGRAPH_6:2;
A7: a<b by A4,A5,XXREAL_0:2;
  then W-min K=|[a,c]| by A1,JGRAPH_6:46;
  then
A8: (W-min(K))`1=a by EUCLID:52;
  p2 in LSeg(|[b,c]|,|[a,c]|) by A4,A5,A7,Th1;
  hence thesis by A1,A4,A7,A6,A8,JGRAPH_6:59;
end;
