reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;

theorem Th18:
  for S being non empty SubSpace of T, s being Point of S,
      f being Loop of t, g being Loop of s st
   t = s & f = g & g is nullhomotopic
  holds f is nullhomotopic
  proof
    let S be non empty SubSpace of T;
    let s be Point of S;
    let f be Loop of t;
    let g be Loop of s such that
A1: t = s & f = g and
A2: g is nullhomotopic;
    consider c be constant Loop of s such that
A3: g,c are_homotopic by A2;
    c = I[01] --> s by BORSUK_2:5 .= I[01] --> t by A1;
    then reconsider c as constant Loop of t by JORDAN:41;
    f,c are_homotopic by A1,A3,Th6;
    hence thesis;
  end;
