reserve T for non empty TopSpace,
  a, b, c, d for Point of T;

theorem Th40:
  for X being non empty TopStruct, a, b being Point of X, P being
Path of a,b st P.0 = a & P.1 = b holds (P * L[01]((0,1)(#),(#)(0,1))).0 = b & (
  P * L[01]((0,1)(#),(#)(0,1))).1 = a
proof
A1: 0 in [. 0,1 .] by XXREAL_1:1;
  set e = L[01]((0,1)(#),(#)(0,1));
  let X be non empty TopStruct, a, b be Point of X;
  let P be Path of a,b such that
A2: P.0 = a and
A3: P.1 = b;
A4: the carrier of Closed-Interval-TSpace(0,1) = [. 0,1 .] by TOPMETR:18;
  e.0 = e.(#)(0,1) by TREAL_1:def 1
    .= (0,1)(#) by TREAL_1:9
    .= 1 by TREAL_1:def 2;
  hence (P * e).0 = b by A3,A4,A1,FUNCT_2:15;
A5: 1 in [. 0,1 .] by XXREAL_1:1;
  e.1 = e.(0,1)(#) by TREAL_1:def 2
    .= (#)(0,1) by TREAL_1:9
    .= 0 by TREAL_1:def 1;
  hence thesis by A2,A4,A5,FUNCT_2:15;
end;
