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,g being Assign of BASSModel(R,BASSIGN) holds SIGMA(f EU g) = lfp(
  S,TransEU(f,g))
proof
  let f,g be Assign of BASSModel(R,BASSIGN);
  set h = f EU g;
  set p = Tau(lfp(S,TransEU(f,g)),R,BASSIGN);
A1: SIGMA(p) = lfp(S,TransEU(f,g)) by Th32;
  lfp(S,TransEU(f,g)) is_a_fixpoint_of TransEU(f,g) by KNASTER:4;
  then
A2: for s being Element of S holds s|= p iff s|= Foax(g,f,p) by A1,Th48;
A3: SIGMA(h) c= SIGMA(p)
  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|= p by A2,A5,Th47;
    hence thesis by A4;
  end;
  for s being Element of S holds s|= h iff s|= Foax(g,f,h) by Th46;
  then SIGMA(h) is_a_fixpoint_of TransEU(f,g) by Th48;
  then lfp(S,TransEU(f,g)) c= SIGMA(h) by KNASTER:8;
  hence thesis by A1,A3,XBOOLE_0:def 10;
end;
