theorem Th1: :: JORDAN5C:1

for P, Q being Subset of (TOP-REAL 2)
for p1, p2, q1 being Point of (TOP-REAL 2)
for f being Function of I[01],((TOP-REAL 2) | P)
for s1 being Real st q1 in P & f . s1 = q1 & f is being_homeomorphism & f . 0 = p1 & f . 1 = p2 & 0 <= s1 & s1 <= 1 & ( for t being Real st 0 <= t & t < s1 holds
not f . t in Q ) holds
for g being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = q1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds
not g . t in Q