
theorem
  for S, T being non empty TopSpace, s being Point of S, t being Point
  of T, f being continuous Function of S,T, P being Path of t,f.s, h being
  Homomorphism of pi_1(S,s),pi_1(T,t) st f is being_homeomorphism & f.s,t
  are_connected & h = pi_1-iso(P) * FundGrIso(f,s) holds h is bijective
proof
  let S, T be non empty TopSpace;
  let s be Point of S;
  let t be Point of T;
  let f be continuous Function of S,T;
  let P be Path of t,f.s;
  let h be Homomorphism of pi_1(S,s),pi_1(T,t);
  assume f is being_homeomorphism;
  then
A1: FundGrIso(f,s) is bijective by Th31;
  assume that
A2: f.s,t are_connected and
A3: h = pi_1-iso(P) * FundGrIso(f,s);
  reconsider G = pi_1-iso(P) as Homomorphism of pi_1(T,f.s),pi_1(T,t) by A2,
TOPALG_1:50;
  G is bijective by A2,TOPALG_1:55;
  hence thesis by A1,A3,GROUP_6:64;
end;
