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 Th29:
  for X being 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 & x1,x2 are_connected &
  rng f c= the carrier of Y holds y1,y2 are_connected & f is Path of y1,y2
proof
  let X be non empty TopSpace, Y be non empty SubSpace of X,
  x1, x2 be Point of X, y1, y2 be Point of Y, f be Path of x1,x2 such that
A1: x1 = y1 and
A2: x2 = y2 and
A3: x1, x2 are_connected;
  assume rng f c= the carrier of Y;
  then reconsider g = f as Function of I[01], Y by FUNCT_2:6;
A4: f is continuous by A3,BORSUK_2:def 2;
A5: f.0 = y1 & f.1 = y2 by A1,A2,A3,BORSUK_2:def 2;
A6: g is continuous by A4,PRE_TOPC:27;
  thus
  ex f being Function of I[01], Y st f is continuous & f.0 = y1 & f.1 = y2
  proof
    take g;
    thus g is continuous by A4,PRE_TOPC:27;
    thus thesis by A1,A2,A3,BORSUK_2:def 2;
  end;
  y1, y2 are_connected by A5,A6;
  hence thesis by A5,A6,BORSUK_2:def 2;
end;
