theorem Th58:
  for f being Function of I[01],TOP-REAL n st f is continuous &
rng f c=P & f.0=p1 & f.1=p2 & p in Ball(u,r) & p2 in Ball(u,r) & Ball(u,r) c= P
ex f1 being Function of I[01],TOP-REAL n st f1 is continuous & rng f1 c= P & f1
  .0=p1 & f1.1=p
proof
  let f be Function of I[01],TOP-REAL n;
  assume f is continuous & rng f c= P & f.0=p1 & f.1=p2 & p in Ball(u,r) & p2
  in Ball(u,r) & Ball(u,r) c= P;
  then
  (ex f3 being Function of I[01],TOP-REAL n st f3 is continuous & f3.0=p1
  & f3.1=p & rng f3 c= rng f \/ Ball(u,r) )& rng f \/ Ball(u,r) c= P by Th57,
XBOOLE_1:8;
  hence thesis by XBOOLE_1:1;
end;
