theorem Th30:
  for X being pathwise_connected non empty TopSpace,
  Y being non empty SubSpace of X,
  x1, x2 being Point of X, y1, y2 being Point of Y,
  f being Path of x1,x2 st x1 = y1 & x2 = y2 & rng f c= the carrier of Y holds
  y1,y2 are_connected & f is Path of y1,y2
proof
  let X be pathwise_connected non empty TopSpace, Y be non empty SubSpace of X,
  x1, x2 be Point of X, y1, y2 be Point of Y;
  x1,x2 are_connected by BORSUK_2:def 3;
  hence thesis by Th29;
end;
