reserve T,T1,T2,S for non empty TopSpace;
reserve GY for non empty TopSpace,
  r,s for Real;

theorem Th13:
  for T being non empty TopSpace, a, b being Point of T st (ex f
  being Function of I[01], T st f is continuous & f.0 = a & f.1 = b) holds ex g
  being Function of I[01], T st g is continuous & g.0 = b & g.1 = a
proof
  set e = L[01]((0,1)(#),(#)(0,1));
  let T be non empty TopSpace, a, b be Point of T;
  given P being Function of I[01], T such that
A1: P is continuous and
A2: P.0 = a & P.1 = b;
  set f = P * e;
  reconsider f as Function of I[01], T by TOPMETR:20;
  take f;
  e is continuous Function of Closed-Interval-TSpace(0,1),
  Closed-Interval-TSpace(0,1) by TREAL_1:8;
  hence f is continuous by A1,TOPMETR:20;
A3: e.1 = e.(0,1)(#) by TREAL_1:def 2
    .= (#)(0,1) by TREAL_1:9
    .= 0 by TREAL_1:def 1;
  1 in [. 0,1 .] by XXREAL_1:1;
  then 1 in the carrier of Closed-Interval-TSpace (0,1) by TOPMETR:18;
  then
A4: 1 in dom e by FUNCT_2:def 1;
  0 in [. 0,1 .] by XXREAL_1:1;
  then 0 in the carrier of Closed-Interval-TSpace (0,1) by TOPMETR:18;
  then
A5: 0 in dom e by FUNCT_2:def 1;
  e.0 = e.(#)(0,1) by TREAL_1:def 1
    .= (0,1)(#) by TREAL_1:9
    .= 1 by TREAL_1:def 2;
  hence thesis by A2,A3,A5,A4,FUNCT_1:13;
end;
