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 Th48:
  for f,g,h being Assign of BASSModel(R,BASSIGN) holds (for s being
  Element of S holds s|= h iff s|= Foax(g,f,h)) iff SIGMA(h) is_a_fixpoint_of
  TransEU(f,g)
proof
  let f,g,h be Assign of BASSModel(R,BASSIGN);
  set H = SIGMA(h);
  set Q = SIGMA(Foax(g,f,h));
A1: (TransEU(f,g)).H = SigFoaxTau(g,f,H,R,BASSIGN) by Def73
    .= Q by Th31;
A2: H is_a_fixpoint_of TransEU(f,g) implies for s being Element of S holds
  s|= h iff s|= Foax(g,f,h)
  proof
    assume H is_a_fixpoint_of TransEU(f,g);
    then
A3: H = Q by A1,ABIAN:def 4;
    for s being Element of S holds s|= h iff s|= Foax(g,f,h)
    proof
      let s be Element of S;
      thus s|= h implies s|= Foax(g,f,h)
      proof
        assume s|= h;
        then s in H;
        then ex t be Element of S st s=t & t|= Foax(g,f,h) by A3;
        hence thesis;
      end;
      assume s|= Foax(g,f,h);
      then s in Q;
      then ex t be Element of S st s =t & t|= h by A3;
      hence thesis;
    end;
    hence thesis;
  end;
  (for s being Element of S holds s|= h iff s|= Foax(g,f,h)) implies H
  is_a_fixpoint_of TransEU(f,g)
  proof
    assume
A4: for s being Element of S holds s|= h iff s|= Foax(g,f,h);
A5: for s be object st s in Q holds s in H
    proof
      let x be object;
      assume x in Q;
      then consider s be Element of S such that
A6:   x=s and
A7:   s|= Foax(g,f,h);
      s|= h by A4,A7;
      hence thesis by A6;
    end;
    for x be object st x in H holds x in Q
    proof
      let x be object;
      assume x in H;
      then consider s be Element of S such that
A8:   x=s and
A9:   s|= h;
      s|= Foax(g,f,h) by A4,A9;
      hence thesis by A8;
    end;
    then H = Q by A5,TARSKI:2;
    hence thesis by A1,ABIAN:def 4;
  end;
  hence thesis by A2;
end;
