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 Th42:
  for P being Subset of TOP-REAL n, p1,p2 being Point of TOP-REAL n st
  P is_an_arc_of p1,p2 ex p3 being Point of TOP-REAL n st
  p3 in P & p3<>p1 & p3<>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;
  1/2 in [#]I[01] by BORSUK_1:40,XXREAL_1:1;
  then
A4: 1/2 in dom f by A1,TOPS_2:def 5;
  then f.(1/2) in rng f by FUNCT_1:def 3;
  then f.(1/2) in the carrier of ((TOP-REAL n)|P);
  then
A5: f.(1/2) in P by PRE_TOPC:8;
  then reconsider p=f.(1/2) as Point of TOP-REAL n;
A6: 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;
  then
A7: p1<>p by A2,A4,A6,FUNCT_1:def 4;
  1 in [#](I[01]) by BORSUK_1:40,XXREAL_1:1;
  then 1 in dom f by A1,TOPS_2:def 5;
  then f.1<>f.(1/2) by A4,A6,FUNCT_1:def 4;
  hence thesis by A3,A5,A7;
end;
