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
  for f being Assign of BASSModel(R,BASSIGN) holds SIGMA(EG(f)) = gfp(S,
  TransEG(f))
proof
  let f be Assign of BASSModel(R,BASSIGN);
  set g = EG(f);
  set h = Tau(gfp(S,TransEG(f)),R,BASSIGN);
A1: SIGMA(h) = gfp(S,TransEG(f)) by Th32;
  then SIGMA(h) is_a_fixpoint_of TransEG(f) by KNASTER:5;
  then
A2: for s being Element of S holds s|= h iff s|= Fax(f,h) by Th42;
A3: SIGMA(h) c= SIGMA(g)
  proof
    let x be object;
    assume x in SIGMA(h);
    then consider s be Element of S such that
A4: x=s and
A5: s|= h;
    s|= g by A2,A5,Th41;
    hence thesis by A4;
  end;
  for s being Element of S holds s|= g iff s|= Fax(f,g) by Th39;
  then SIGMA(g) is_a_fixpoint_of TransEG(f) by Th42;
  then SIGMA(g) c= gfp(S,TransEG(f)) by KNASTER:8;
  hence thesis by A1,A3,XBOOLE_0:def 10;
end;
