reserve k,n for Element of NAT,
  a,Y for set,
  D,D1,D2 for non empty set,
  p,q for FinSequence of NAT;
reserve F,F1,G,G1,H,H1,H2 for CTL-formula;
reserve sq,sq9 for FinSequence;
reserve V for CTLModel;
reserve Kai for Function of atomic_WFF,the BasicAssign of V;
reserve f,f1,f2 for Function of CTL_WFF,the carrier of V;
reserve S for non empty set;
reserve R for total Relation of S,S;
reserve s,s0,s1 for Element of S;
reserve BASSIGN for non empty Subset of ModelSP(S);
reserve kai for Function of atomic_WFF,the BasicAssign of BASSModel(R,BASSIGN);

theorem Th42:
  for f,g being Assign of BASSModel(R,BASSIGN) holds (for s being
  Element of S holds s|= g iff s|= Fax(f,g)) iff SIGMA(g) is_a_fixpoint_of
  TransEG(f)
proof
  let f,g be Assign of BASSModel(R,BASSIGN);
  set G = SIGMA(g);
  set Q = SIGMA(Fax(f,g));
A1: (TransEG(f)).G = SigFaxTau(f,G,R,BASSIGN) by Def70
    .= Q by Th31;
A2: G is_a_fixpoint_of TransEG(f) implies for s being Element of S holds s
  |= g iff s|= Fax(f,g)
  proof
    assume G is_a_fixpoint_of TransEG(f);
    then
A3: G = Q by A1,ABIAN:def 4;
    for s being Element of S holds s|= g iff s|= Fax(f,g)
    proof
      let s be Element of S;
      thus s|= g implies s|= Fax(f,g)
      proof
        assume s|= g;
        then s in Q by A3;
        then ex t be Element of S st s=t & t|= Fax(f,g);
        hence thesis;
      end;
      assume s|= Fax(f,g);
      then s in G by A3;
      then ex t be Element of S st s = t & t|= g;
      hence thesis;
    end;
    hence thesis;
  end;
  (for s being Element of S holds s|= g iff s|= Fax(f,g)) implies G
  is_a_fixpoint_of TransEG(f)
  proof
    assume
A4: for s being Element of S holds s|= g iff s|= Fax(f,g);
A5: for s be object st s in Q holds s in G
    proof
      let x be object;
      assume x in Q;
      then consider s be Element of S such that
A6:   x=s and
A7:   s|= Fax(f,g);
      s|= g by A4,A7;
      hence thesis by A6;
    end;
    for x be object st x in G holds x in Q
    proof
      let x be object;
      assume x in G;
      then consider s be Element of S such that
A8:   x=s and
A9:   s|= g;
      s|= Fax(f,g) by A4,A9;
      hence thesis by A8;
    end;
    then G = (TransEG(f)).G by A1,A5,TARSKI:2;
    hence thesis by ABIAN:def 4;
  end;
  hence thesis by A2;
end;
