
theorem
  for p1,p2,p3,p4 being Point of TOP-REAL 2, a,b,c,d being Real,
P,Q being Subset of TOP-REAL 2 st a<b & c < d & p1`2=d & p2`1=b & p3`1=b & p4`2
= c & a <=p1`1 & p1`1<=b & d>=p2`2 & p2`2>p3`2 & p3`2>= c & a <p4`1 & p4`1<=b &
P is_an_arc_of p1,p3 & Q is_an_arc_of p2,p4 & P c= closed_inside_of_rectangle(a
  ,b,c,d) & Q c= closed_inside_of_rectangle(a,b,c,d) holds P meets Q
proof
  let p1,p2,p3,p4 be Point of TOP-REAL 2, a,b,c,d be Real, P,Q be
  Subset of TOP-REAL 2;
  assume that
A1: a<b and
A2: c < d and
A3: p1`2=d and
A4: p2`1=b and
A5: p3`1=b and
A6: p4`2= c and
A7: a <=p1`1 and
A8: p1`1<=b and
A9: d>=p2`2 and
A10: p2`2>p3`2 and
A11: p3`2>= c and
A12: a <p4`1 and
A13: p4`1<=b and
A14: P is_an_arc_of p1,p3 and
A15: Q is_an_arc_of p2,p4 and
A16: P c= closed_inside_of_rectangle(a,b,c,d) and
A17: Q c= closed_inside_of_rectangle(a,b,c,d);
A18: ex g being Function of I[01],TOP-REAL 2 st g is continuous one-to-one
  & rng g=Q & g.0=p2 & g.1=p4 by A15,Th2;
  ex f being Function of I[01],TOP-REAL 2 st f is continuous one-to-one
  & rng f=P & f.0=p1 & f.1=p3 by A14,Th2;
  hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A16,A17,A18,Th122;
end;
