
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`1=a & p2`1=a & p3`2= c & p4
  `2= c & c <=p1`2 & p1`2<p2`2 & p2`2<=d & a<p4`1 & p4`1<p3`1 & p3`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`1=a and
A4: p2`1=a and
A5: p3`2= c and
A6: p4`2= c and
A7: c <=p1`2 and
A8: p1`2<p2`2 and
A9: p2`2<=d and
A10: a<p4`1 and
A11: p4`1<p3`1 and
A12: p3`1<=b and
A13: P is_an_arc_of p1,p3 and
A14: Q is_an_arc_of p2,p4 and
A15: P c= closed_inside_of_rectangle(a,b,c,d) and
A16: Q c= closed_inside_of_rectangle(a,b,c,d);
A17: 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 A14,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 A13,Th2;
  hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A15,A16,A17,Th86;
end;
