reserve p,p1,p2,p3,q for Point of TOP-REAL 2;
reserve n for Nat;

theorem Th17:
  for A being Subset of TOP-REAL 2, p1,p2 being Point of TOP-REAL
  2 st A is_an_arc_of p1,p2 ex g being Function of I[01], TOP-REAL 2 st g is
  continuous one-to-one & rng g = A & g.0 = p1 & g.1 = p2
proof
  let A be Subset of TOP-REAL 2, p1,p2 being Point of TOP-REAL 2;
  assume
A1: A is_an_arc_of p1,p2;
  then reconsider A9 = A as non empty Subset of TOP-REAL 2 by TOPREAL1:1;
  consider f being Function of I[01], (TOP-REAL 2)|A9 such that
A2: f is being_homeomorphism and
A3: f.0 = p1 & f.1 = p2 by A1,TOPREAL1:def 1;
  consider g being Function of I[01], TOP-REAL 2 such that
A4: f=g and
A5: g is continuous one-to-one by A2,Th15;
  take g;
  thus g is continuous one-to-one by A5;
  rng f=[#]((TOP-REAL 2)|A9) by A2,TOPS_2:def 5;
  hence rng g = A by A4,PRE_TOPC:def 5;
  thus thesis by A3,A4;
end;
