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

theorem Th6:
  for S being non empty SubSpace of T,
      t1,t2 being Point of T, s1,s2 being Point of S,
      A,B being Path of t1,t2, C,D being Path of s1,s2 st
  s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & C,D are_homotopic
  holds A,B are_homotopic
  proof
    let S be non empty SubSpace of T;
    let t1,t2 be Point of T;
    let s1,s2 be Point of S;
    let A,B be Path of t1,t2;
    let C,D be Path of s1,s2 such that
A1: s1,s2 are_connected & t1,t2 are_connected and
A2: A = C & B = D;
    given f being Function of [:I[01],I[01]:],S such that
A3: f is continuous and
A4: for t being Point of I[01] holds f.(t,0) = C.t & f.(t,1) = D.t &
    f.(0,t) = s1 & f.(1,t) = s2;
    reconsider g = f as Function of [:I[01],I[01]:],T by TOPREALA:7;
    take g;
    thus g is continuous by A3,PRE_TOPC:26;
    s1 = C.0 & s2 = C.1 & t1 = A.0 & t2 = A.1 by A1,BORSUK_2:def 2;
    hence thesis by A2,A4;
  end;
