
theorem
  for S, T being non empty TopSpace, s being Point of S, f being
continuous Function of S,T, ls being Loop of s holds FundGrIso(f,s).Class(EqRel
  (S,s),ls) = Class(EqRel(T,f.s),f*ls)
proof
  let S, T be non empty TopSpace;
  let s be Point of S;
  let f be continuous Function of S,T;
  let ls be Loop of s;
  reconsider x = Class(EqRel(S,s),ls) as Element of pi_1(S,s) by TOPALG_1:47;
  consider ls1 being Loop of s such that
A1: x = Class(EqRel(S,s),ls1) and
A2: FundGrIso(f,s).x = Class(EqRel(T,f.s),f*ls1) by Def1;
  ls,ls1 are_homotopic by A1,TOPALG_1:46;
  then f*ls,f*ls1 are_homotopic by Th27;
  hence thesis by A2,TOPALG_1:46;
end;
