theorem Th58: :: JORDAN2C:74

for n being Nat
for r being Real
for p, p1, p2 being Point of (TOP-REAL n)
for u being Point of (Euclid n)
for P being Subset of (TOP-REAL n)
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 holds
ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p )