
theorem Th2:
  for n being Element of NAT, P being Subset of TOP-REAL n, p1,p2
being Point of TOP-REAL n st P is_an_arc_of p1,p2 holds ex f being Function of
  I[01],TOP-REAL n st f is continuous one-to-one & rng f=P & f.0=p1 & f.1=p2
proof
  let n be Element of NAT,P be Subset of TOP-REAL n, p1,p2 be Point of
  TOP-REAL n;
  assume
A1: P is_an_arc_of p1,p2;
  then consider f2 being Function of I[01], (TOP-REAL n)|P such that
A2: f2 is being_homeomorphism and
A3: f2.0 = p1 and
A4: f2.1 = p2 by TOPREAL1:def 1;
  p1 in P by A1,TOPREAL1:1;
  then consider g being Function of I[01],TOP-REAL n such that
A5: f2=g and
A6: g is continuous and
A7: g is one-to-one by A2,JORDAN7:15;
  rng g =[#]((TOP-REAL n)|P) by A2,A5,TOPS_2:def 5
    .=P by PRE_TOPC:def 5;
  hence thesis by A3,A4,A5,A6,A7;
end;
