
theorem Th31:
  for S, T being non empty TopSpace, s being Point of S, f being
continuous Function of S,T st f is being_homeomorphism holds FundGrIso(f,s) is
  bijective
proof
  let S, T be non empty TopSpace;
  let s be Point of S;
  set pS = pi_1(S,s);
  let f be continuous Function of S,T such that
A1: f is being_homeomorphism;
A2: f is one-to-one by A1;
  then
A3: f qua Function".(f.s) = s by FUNCT_2:26;
  set h = FundGrIso(f,s);
  set pT = pi_1(T,f.s);
A4: f" is continuous by A1;
A5: rng f = [#]T by A1;
  then f is onto;
  then
A6: f qua Function" = f" by A2,TOPS_2:def 4;
A7: dom h = the carrier of pS by FUNCT_2:def 1;
A8: rng h = the carrier of pT
  proof
    thus rng h c= the carrier of pT;
    let y be object;
    assume y in the carrier of pT;
    then consider lt being Loop of f.s such that
A9: y = Class(EqRel(T,f.s),lt) by TOPALG_1:47;
    reconsider ls = f"*lt as Loop of s by A4,A3,A6,Th26;
    set x = Class(EqRel(S,s),ls);
A10: x in the carrier of pS by TOPALG_1:47;
    then consider ls1 being Loop of s such that
A11: x = Class(EqRel(S,s),ls1) and
A12: h.x = Class(EqRel(T,f.s),f*ls1) by Def1;
A13: f*ls = (f*f")*lt by RELAT_1:36
      .= id rng f * lt by A5,A2,TOPS_2:52
      .= lt by A5,FUNCT_2:17;
    ls,ls1 are_homotopic by A11,TOPALG_1:46;
    then lt,f*ls1 are_homotopic by A13,Th27;
    then h.x = y by A9,A12,TOPALG_1:46;
    hence thesis by A7,A10,FUNCT_1:def 3;
  end;
  h is one-to-one
  proof
    let x1, x2 be object;
    assume x1 in dom h;
    then consider ls1 being Loop of s such that
A14: x1 = Class(EqRel(S,s),ls1) and
A15: h.x1 = Class(EqRel(T,f.s),f*ls1) by Def1;
    assume x2 in dom h;
    then consider ls2 being Loop of s such that
A16: x2 = Class(EqRel(S,s),ls2) and
A17: h.x2 = Class(EqRel(T,f.s),f*ls2) by Def1;
    reconsider a1 = f"*(f*ls1), a2 = f"*(f*ls2) as Loop of s by A4,A3,A6,Th26;
    assume h.x1 = h.x2;
    then f*ls1,f*ls2 are_homotopic by A15,A17,TOPALG_1:46;
    then
A18: a1,a2 are_homotopic by A4,A3,A6,Th27;
A19: f"*f = id dom f by A5,A2,TOPS_2:52;
A20: f"*(f*ls1) = (f"*f)*ls1 by RELAT_1:36
      .= ls1 by A19,FUNCT_2:17;
    f"*(f*ls2) = (f"*f)*ls2 by RELAT_1:36
      .= ls2 by A19,FUNCT_2:17;
    hence thesis by A14,A16,A20,A18,TOPALG_1:46;
  end;
  hence thesis by A8,GROUP_6:60;
end;
