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 Th31:
  for T being non empty TopSpace, a, b being Point of T
  for f being Path of a,b st a,b are_connected holds rng f = rng -f
proof
  let T be non empty TopSpace;
  let a, b be Point of T;
  let f be Path of a,b;
  assume
A1: a,b are_connected;
  hence rng f c= rng -f by Lm7;
  f = --f by A1,BORSUK_6:43;
  hence thesis by A1,Lm7;
end;
