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 Th16:
  s1,s2 are_connected & t1,t2 are_connected implies for L1 being
Path of s1,s2, L2 being Path of t1,t2 holds <:L1,L2:> is Path of [s1,t1],[s2,t2
  ]
proof
  assume that
A1: s1,s2 are_connected and
A2: t1,t2 are_connected;
  let L1 be Path of s1,s2, L2 be Path of t1,t2;
  L1 is continuous & L2 is continuous by A1,A2,BORSUK_2:def 2;
  then
A3: <:L1,L2:> is continuous by YELLOW12:41;
A4: dom L1 = I & dom L2 = I by FUNCT_2:def 1;
  then
A5: <:L1,L2:>.j1 = [L1.j1,L2.j1] by FUNCT_3:49
    .= [s2,L2.j1] by A1,BORSUK_2:def 2
    .= [s2,t2] by A2,BORSUK_2:def 2;
A6: <:L1,L2:>.j0 = [L1.j0,L2.j0] by A4,FUNCT_3:49
    .= [s1,L2.j0] by A1,BORSUK_2:def 2
    .= [s1,t1] by A2,BORSUK_2:def 2;
  then [s1,t1], [s2,t2] are_connected by A3,A5;
  hence thesis by A3,A6,A5,BORSUK_2:def 2;
end;
