
theorem Th12:
  for p1,p2 being Point of TOP-REAL 2, a,b,c,d being Real
st a<b & c <d & p1`2= c & p2`2= c & a <p2`1 & p2`1<p1`1 & p1`1<=b holds LE p1,
  p2,rectangle(a,b,c,d)
proof
  let p1,p2 be Point of TOP-REAL 2, a,b,c,d be 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: a <p2`1 and
A6: p2`1<p1`1 and
A7: p1`1<=b;
  b>p2`1 by A6,A7,XXREAL_0:2;
  then
A8: p2 in LSeg(|[b,c]|,|[a,c]|) by A1,A4,A5,Th1;
  W-min K=|[a,c]| by A1,A2,JGRAPH_6:46;
  then
A9: (W-min(K))`1=a by EUCLID:52;
  p1`1>a by A5,A6,XXREAL_0:2;
  then p1 in LSeg(|[b,c]|,|[a,c]|) by A1,A3,A7,Th1;
  hence thesis by A1,A2,A5,A6,A8,A9,JGRAPH_6:62;
end;
