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 Th22:
  for p, q being Path of s1,s2, x, y being Path of t1, t2 st p =
  pr1 l1 & q = pr1 l2 & x = pr2 l1 & y = pr2 l2 & p,q are_homotopic & x,y
  are_homotopic holds l1, l2 are_homotopic
proof
  let p, q be Path of s1,s2, x, y be Path of t1, t2 such that
A1: p = pr1 l1 & q = pr1 l2 and
A2: x = pr2 l1 & y = pr2 l2 and
A3: p,q are_homotopic and
A4: x,y are_homotopic;
  consider g being Function of [:I[01],I[01]:], T such that
A5: g is continuous & for a being Point of I[01] holds g.(a,0) = pr2 l1.
a & g.(a,1) = pr2 l2.a & for b being Point of I[01] holds g.(0,b) = t1 & g.(1,
  b) = t2 by A2,A4;
A6: g is Homotopy of x,y by A2,A4,A5,BORSUK_6:def 11;
  consider f being Function of [:I[01],I[01]:], S such that
A7: f is continuous & for a being Point of I[01] holds f.(a,0) = pr1 l1.
a & f.(a,1) = pr1 l2.a & for b being Point of I[01] holds f.(0,b) = s1 & f.(1,
  b) = s2 by A1,A3;
  take <:f,g:>;
  f is Homotopy of p,q by A1,A3,A7,BORSUK_6:def 11;
  hence thesis by A1,A2,A3,A4,A6,Lm5;
end;
