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

theorem Th39:
  for P being Path of a,b st P is continuous holds P * L[01]((0,1)
  (#),(#)(0,1)) is continuous Function of I[01], T
proof
  reconsider g = L[01]((0,1)(#),(#)(0,1)) as Function of I[01], I[01] by
TOPMETR:20;
  let P be Path of a,b such that
A1: P is continuous;
  reconsider f = P * g as Function of I[01], T;
  g is continuous by TOPMETR:20,TREAL_1:8;
  then f is continuous by A1;
  hence thesis;
end;
