reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

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;
