theorem Th52: :: BORSUK_4:55

for Dp being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | Dp),I(01)
for C being non empty Subset of (TOP-REAL 2) st f is being_homeomorphism & C c= Dp & ex p1, p2 being Point of I[01] st
( p1 < p2 & f .: C = [.p1,p2.] ) holds
ex s1, s2 being Point of (TOP-REAL 2) st C is_an_arc_of s1,s2