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

theorem Th16:
  for f being Loop of t, g being continuous Function of T,U st
   f is nullhomotopic holds g*f is nullhomotopic
  proof
    let f be Loop of t, g be continuous Function of T,U;
    given c being constant Loop of t such that
A1: f,c are_homotopic;
    consider F being Function of [:I[01],I[01]:], T such that
A2: F is continuous and
A3: for s being Point of I[01] holds F.(s,0) = f.s & F.(s,1) = c.s &
    F.(0,s) = t & F.(1,s) = t by A1;
    reconsider d = I[01] --> g.t as constant Loop of g.t by JORDAN:41;
    reconsider G = g*F as Function of [:I[01],I[01]:], U;
    take d, G;
    thus G is continuous by A2;
    let s be Point of I[01];
    reconsider j0 = 0, j1 = 1 as Point of I[01] by BORSUK_1:def 14,def 15;
    set I = the carrier of I[01];
A4: the carrier of [:I[01],I[01]:] = [:I,I:] by BORSUK_1:def 2;
    thus G.(s,0) = g.(F.(s,j0)) by A4,BINOP_1:18
    .= g.(f.s) by A3
    .= (g*f).s by FUNCT_2:15;
    thus G.(s,1) = g.(F.(s,j1)) by A4,BINOP_1:18
    .= g.(c.s) by A3
    .= g.t by TOPALG_3:21
    .= d.s;
    thus G.(0,s) = g.(F.(j0,s)) by A4,BINOP_1:18
    .= g.t by A3;
    thus G.(1,s) = g.(F.(j1,s)) by A4,BINOP_1:18
    .= g.t by A3;
  end;
