
theorem Th3:
  for p1,p2 being Point of TOP-REAL 2, b,c,d being Real st
p1`1<b & p1`1=p2`1 & c <=p1`2 & p1`2<p2`2 & p2`2<=d holds LE p1,p2,rectangle(p1
  `1,b,c,d)
proof
  let p1,p2 be Point of TOP-REAL 2,b,c,d be Real;
  set a =p1`1;
  assume that
A1: p1`1<b and
A2: p1`1=p2`1 and
A3: c <=p1`2 and
A4: p1`2<p2`2 and
A5: p2`2<=d;
A6: p1`2< d by A4,A5,XXREAL_0:2;
  then
A7: c < d by A3,XXREAL_0:2;
  then
A8: p1 in LSeg(|[a,c]|,|[a,d]|) by A3,A6,JGRAPH_6:2;
  c <=p2`2 by A3,A4,XXREAL_0:2;
  then p2 in LSeg(|[a,c]|,|[a,d]|) by A2,A5,A7,JGRAPH_6:2;
  hence thesis by A1,A4,A7,A8,JGRAPH_6:55;
end;
