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 Th39:
  for T being non empty TopSpace, a, b, c, d being Point of T
  for f being Path of a,b, g being Path of b,c, h being Path of c,d st
  a,b are_connected & b,c are_connected & c,d are_connected holds
  rng(f+g+h) = rng f \/ rng g \/ rng h
proof
  let T be non empty TopSpace;
  let a, b, c, d be Point of T;
  let f be Path of a,b;
  let g be Path of b,c;
  let h be Path of c,d;
  assume that
A1: a,b are_connected and
A2: b,c are_connected and
A3: c,d are_connected;
  a,c are_connected by A1,A2,BORSUK_6:42;
  hence rng(f+g+h) = rng(f+g) \/ rng h by A3,Th37
    .= rng f \/ rng g \/ rng h by A1,A2,Th37;
end;
