reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;
reserve R for Subset of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL n;

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;
