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 Th43:
  for p1, p2 being Point of TOP-REAL n
  ex F being Path of p1,p2, f being Function
  of I[01], (TOP-REAL n)|LSeg(p1,p2) st rng f = LSeg(p1,p2) & F = f
proof
  let p1, p2 be Point of TOP-REAL n;
  per cases;
  suppose
A1: p1 = p2;
    then reconsider g = I[01] --> p1 as Path of p1,p2 by Th41;
    take g;
A2: LSeg(p1,p2) = {p1} by A1,RLTOPSP1:70;
A3: rng g = {p1} by FUNCOP_1:8;
    the carrier of (TOP-REAL n)|LSeg(p1,p2) = LSeg(p1,p2) by PRE_TOPC:8;
    then reconsider f = g as Function of I[01],(TOP-REAL n)|LSeg(p1,p2)
    by A2,A3,FUNCT_2:6;
    take f;
    thus thesis by A1,A3,RLTOPSP1:70;
  end;
  suppose p1 <> p2;
    hence thesis by Th42,TOPREAL1:9;
  end;
end;
