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 Th14:
  [s1,t1],[s2,t2] are_connected implies for L being Path of [s1,t1
  ],[s2,t2] holds pr2 L is Path of t1,t2
proof
  assume
A1: [s1,t1], [s2,t2] are_connected;
  let L be Path of [s1,t1],[s2,t2];
  set f = pr2 L;
A2: dom f = I & dom f = dom L by FUNCT_2:def 1,MCART_1:def 13;
  then j0 in dom L;
  then
A3: f.0 = (L.0)`2 by MCART_1:def 13
    .= [s1,t1]`2 by A1,BORSUK_2:def 2
    .= t1;
  j1 in dom L by A2;
  then
A4: f.1 = (L.1)`2 by MCART_1:def 13
    .= [s2,t2]`2 by A1,BORSUK_2:def 2
    .= t2;
  L is continuous by A1,BORSUK_2:def 2;
  then
A5: f is continuous by Th10;
  then t1,t2 are_connected by A3,A4;
  hence thesis by A5,A3,A4,BORSUK_2:def 2;
end;
