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 st p = pr1 l1 & q = pr1 l2 & l1,l2
  are_homotopic holds pr1 H is Homotopy of p,q
proof
  let p, q be Path of s1,s2 such that
A1: p = pr1 l1 & q = pr1 l2 & l1,l2 are_homotopic;
  thus p,q are_homotopic
  proof
    take pr1 H;
    thus thesis by A1,Lm3;
  end;
  thus thesis by A1,Lm3;
end;
