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
  for p, q being Path of s1,s2, x, y being Path of t1,t2, f being
  Homotopy of p,q, g being Homotopy of x,y st p = pr1 l1 & q = pr1 l2 & x = pr2
  l1 & y = pr2 l2 & p,q are_homotopic & x,y are_homotopic holds <:f,g:> is
  Homotopy of l1,l2
proof
  let p, q be Path of s1,s2, x, y be Path of t1,t2, f be Homotopy of p,q, g be
  Homotopy of x,y such that
A1: p = pr1 l1 & q = pr1 l2 & x = pr2 l1 & y = pr2 l2 & p,q
  are_homotopic & x,y are_homotopic;
  thus l1,l2 are_homotopic
  proof
    take <:f,g:>;
    thus thesis by A1,Lm5;
  end;
  thus thesis by A1,Lm5;
end;
