reserve n for Element of NAT,
  a, b for Real;

theorem Th12:
  for T being non empty interval SubSpace of R^1, a, b being Point
  of T, P, Q being Path of a,b holds P, Q are_homotopic
proof
  let T be non empty interval SubSpace of R^1, a, b be Point of T, P, Q be Path
  of a,b;
  take F = R1Homotopy(P,Q);
  thus F is continuous by Lm8;
  thus thesis by Lm9;
end;
