reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem Th37:
  for P being Subset of TOP-REAL n,
  p1,p2 being Point of TOP-REAL n st P is_an_arc_of p1,p2 holds p1<>p2
proof
  let P be Subset of TOP-REAL n, p1,p2 be Point of TOP-REAL n;
  assume P is_an_arc_of p1,p2;
  then consider f being Function of I[01], (TOP-REAL n)|P such that
A1: f is being_homeomorphism and
A2: f.0 = p1 and
A3: f.1 = p2 by TOPREAL1:def 1;
  1 in [#](I[01]) by BORSUK_1:40,XXREAL_1:1;
  then
A4: 1 in dom f by A1,TOPS_2:def 5;
A5: f is one-to-one by A1,TOPS_2:def 5;
  0 in [#](I[01]) by BORSUK_1:40,XXREAL_1:1;
  then 0 in dom f by A1,TOPS_2:def 5;
  hence thesis by A2,A3,A4,A5,FUNCT_1:def 4;
end;
