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

theorem Th41:
  for P being Path of a,b st P is continuous & P.0 = a & P.1 = b
  holds -P is continuous & (-P).0 = b & (-P).1 = a
proof
  let P be Path of a,b such that
A1: P is continuous and
A2: P.0 = a & P.1 = b;
A3: (P * L[01]((0,1)(#),(#)(0,1))).1 = a by A2,Th40;
  P * L[01]((0,1)(#),(#)(0,1)) is continuous Function of I[01], T & (P *
  L[01] ((0,1)(#),(#)(0,1))).0 = b by A1,A2,Th39,Th40;
  then b,a are_connected by A3;
  hence thesis by BORSUK_2:def 2;
end;
