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

theorem Th13:
  for S, T being non empty TopSpace st S,T are_homeomorphic holds
  S is having_trivial_Fundamental_Group implies
  T is having_trivial_Fundamental_Group
  proof
    let S, T be non empty TopSpace;
    given f being Function of S,T such that
A1: f is being_homeomorphism;
    assume
A2: for s being Point of S holds pi_1(S,s) is trivial;
    let t be Point of T;
    rng f = [#]T by A1,TOPS_2:def 5;
    then consider s being Point of S such that
A3: f.s = t by FUNCT_2:113;
A4: FundGrIso(f,s) is bijective by A1,TOPALG_3:31;
    pi_1(S,s) is trivial by A2;
    hence thesis by A3,A4,KNASTER:11,TOPREALC:1;
  end;
