
theorem Th39:
  for E being Subset of I(01) st (ex p1, p2 being Point of I[01]
  st p1 < p2 & E = [. p1,p2 .]) holds I[01], I(01)|E are_homeomorphic
proof
  let 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: p2 <= 1 by BORSUK_1:43;
  0 <= p1 by BORSUK_1:43;
  then reconsider
  S = Closed-Interval-TSpace(p1,p2) as SubSpace of I[01] by A1,A3,TOPMETR:20
,TREAL_1:3;
  reconsider T = I(01)|E as SubSpace of I[01] by TSEP_1:7;
  the carrier of S = E by A1,A2,TOPMETR:18;
  then
A4: S = T by PRE_TOPC:8,TSEP_1:5;
  set f = L[01]((#)(p1,p2), (p1,p2)(#));
  f is being_homeomorphism by A1,TREAL_1:17;
  hence thesis by A4,TOPMETR:20,T_0TOPSP:def 1;
end;
