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 non empty TopSpace, a, b being Point of T
  for f being Path of a,b st a,b are_connected holds rng f is connected
proof
  let T be non empty TopSpace, a, b be Point of T;
  let f be Path of a,b such that
A1: a,b are_connected;
A2: dom f = the carrier of I[01] by FUNCT_2:def 1;
  reconsider A = [.0,1.] as interval Subset of R^1 by TOPMETR:17;
  reconsider B = A as Subset of I[01] by BORSUK_1:40;
A3: B is connected by CONNSP_1:23;
A4: f is continuous by A1,BORSUK_2:def 2;
  f.:B = rng f by A2,BORSUK_1:40,RELAT_1:113;
  hence thesis by A3,A4,TOPS_2:61;
end;
