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 Th33:
  for T being non empty TopSpace, a, b, c being Point of T
  for f being Path of a,b, g being Path of b,c st
  a,b are_connected & b,c are_connected holds rng f c= rng(f+g)
proof
  let T be non empty TopSpace;
  let a, b, c be Point of T;
  let f be Path of a,b;
  let g be Path of b,c;
  assume that
A1: a,b are_connected and
A2: b,c are_connected;
  let y be object;
  assume y in rng f;
  then consider x being object such that
A3: x in dom f and
A4: f.x = y by FUNCT_1:def 3;
A5: dom(f+g) = the carrier of I[01] by FUNCT_2:def 1;
  reconsider x as Point of I[01] by A3;
  1/2*x = x/2;
  then
A6: x/2 is Point of I[01] by BORSUK_6:6;
  x <= 1 by BORSUK_1:43;
  then x/2 <= 1/2 by XREAL_1:72;
  then (f+g).(x/2) = f.(2*(x/2)) by A1,A2,A6,BORSUK_2:def 5;
  hence thesis by A4,A5,A6,FUNCT_1:def 3;
end;
