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 Th27:
  for x being Point of pi_1([:S,T:],[s,t]), l being Loop of [s,t]
st x = Class(EqRel([:S,T:],[s,t]),l) holds FGPrIso(s,t).x = <*Class(EqRel(S,s),
  pr1 l),Class(EqRel(T,t),pr2 l)*>
proof
  let x be Point of pi_1([:S,T:],[s,t]), l be Loop of [s,t];
  consider l1 being Loop of [s,t] such that
A1: x = Class(EqRel([:S,T:],[s,t]),l1) and
A2: FGPrIso(s,t).x = <*Class(EqRel(S,s),pr1 l1),Class(EqRel(T,t),pr2 l1)
  *> by Def2;
  assume x = Class(EqRel([:S,T:],[s,t]),l);
  then
A3: l,l1 are_homotopic by A1,TOPALG_1:46;
  then pr2 l,pr2 l1 are_homotopic by Th20;
  then
A4: Class(EqRel(T,t),pr2 l) = Class(EqRel(T,t),pr2 l1) by TOPALG_1:46;
  pr1 l,pr1 l1 are_homotopic by A3,Th19;
  hence thesis by A2,A4,TOPALG_1:46;
end;
