reserve S, T, Y for non empty TopSpace,
  s, s1, s2, s3 for Point of S,
  t, t1, t2, t3 for Point of T,
  l1, l2 for Path of [s1,t1],[s2,t2],
  H for Homotopy of l1 ,l2;

theorem Th20:
  for p, q being Path of t1,t2 st p = pr2 l1 & q = pr2 l2 & l1,l2
  are_homotopic holds p,q are_homotopic
proof
  let p, q be Path of t1,t2 such that
A1: p = pr2 l1 & q = pr2 l2 and
A2: l1,l2 are_homotopic;
  consider f being Function of [:I[01],I[01]:], [:S,T:] such that
A3: f is continuous & for a being Point of I[01] holds f.(a,0) = l1.a &
f.(a,1) = l2.a & for b being Point of I[01] holds f.(0,b) = [s1,t1] & f.(1,b) =
  [s2,t2] by A2;
  take pr2 f;
  f is Homotopy of l1,l2 by A2,A3,BORSUK_6:def 11;
  hence thesis by A1,A2,Lm4;
end;
