reserve p,p1,p2,p3,q,q1,q2 for Point of TOP-REAL 2,
  i for Nat,
  lambda for Real;

theorem Th19:
  for a,b,c,d being Real, p being Point of TOP-REAL 2
  st a<=b & c <=d & p in rectangle(a,b,c,d) holds
  a<=p`1 & p`1<=b & c <=p`2 & p`2<=d
proof
  let a,b,c,d be Real, p be Point of TOP-REAL 2;
  assume that
A1: a<=b and
A2: c <=d and
A3: p in rectangle(a,b,c,d);
  p in {p2: p2`1 = a & p2`2 <= d & p2`2 >= c or
  p2`1 <= b & p2`1 >= a & p2`2 = d or p2`1 <= b & p2`1 >= a & p2`2 = c or
  p2`1 = b & p2`2 <= d & p2`2 >= c} by A1,A2,A3,SPPOL_2:54;
  then
A4: ex p2 st ( p2=p)&( p2`1 = a & p2`2 <= d & p2`2 >= c or p2`1
  <= b & p2`1 >= a & p2`2 = d or p2`1 <= b & p2`1 >= a & p2`2 = c or p2`1 = b &
  p2`2 <= d & p2`2 >= c);
  per cases by A4;
  suppose p`1=a & c <=p`2 & p`2<=d;
    hence thesis by A1;
  end;
  suppose p`2=d & a<=p`1 & p`1<=b;
    hence thesis by A2;
  end;
  suppose p`1=b & c <=p`2 & p`2<=d;
    hence thesis by A1;
  end;
  suppose p`2=c & a<=p`1 & p`1<=b;
    hence thesis by A2;
  end;
end;
