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
  for T being pathwise_connected non empty TopSpace, a, b, c being Point of T
  for f being Path of b,c, g being Path of a,b holds rng f c= rng(g+f)
proof
  let T be pathwise_connected non empty TopSpace;
  let a, b, c be Point of T;
  let f be Path of b,c;
  let g be Path of a,b;
A1: a,b are_connected by BORSUK_2:def 3;
  b,c are_connected by BORSUK_2:def 3;
  hence thesis by A1,Th35;
end;
