
theorem Th51:
  for n being Element of NAT, C being non empty Subset of TOP-REAL
n, E being Subset of I(01) st (ex p1, p2 being Point of I[01] st p1 < p2 & E =
  [. p1,p2 .]) & I(01)|E, (TOP-REAL n)|C are_homeomorphic holds ex s1, s2 being
  Point of TOP-REAL n st C is_an_arc_of s1,s2
proof
  let n be Element of NAT, C be non empty Subset of TOP-REAL n, E be Subset of
  I(01);
  given p1, p2 being Point of I[01] such that
A1: p1 < p2 and
A2: E = [. p1,p2 .];
A3: I[01], I(01)|E are_homeomorphic by A1,A2,Th39;
  assume
A4: I(01)|E, (TOP-REAL n)|C are_homeomorphic;
  E is non empty by A1,A2,Th21;
  then I[01], (TOP-REAL n)|C are_homeomorphic by A4,A3,BORSUK_3:3;
  then consider g being Function of I[01], (TOP-REAL n)|C such that
A5: g is being_homeomorphism by T_0TOPSP:def 1;
  set s1 = g.0, s2 = g.1;
  0 in the carrier of I[01] by BORSUK_1:43;
  then
A6: g.0 in the carrier of (TOP-REAL n)|C by FUNCT_2:5;
  1 in the carrier of I[01] by BORSUK_1:43;
  then
A7: g.1 in the carrier of (TOP-REAL n)|C by FUNCT_2:5;
  the carrier of (TOP-REAL n)|C c= the carrier of TOP-REAL n by BORSUK_1:1;
  then reconsider s1, s2 as Point of TOP-REAL n by A6,A7;
  C is_an_arc_of s1, s2 by A5,TOPREAL1:def 1;
  hence thesis;
end;
