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 Th50:
  for f being Assign of BASSModel(R,BASSIGN) holds SIGMA(EX(f)) =
  Pred(SIGMA(f),R)
proof
  let f be Assign of BASSModel(R,BASSIGN);
  set g = EX(f);
  set H = SIGMA(f);
A1: for x being object holds x in Pred(H,R) implies x in SIGMA(g)
  proof
    let x be object;
    assume x in Pred(H,R);
    then consider s be Element of S such that
A2: x = s and
A3: ex s1 being Element of S st s1 in H & [s,s1] in R;
    consider s1 be Element of S such that
A4: s1 in H and
A5: [s,s1] in R by A3;
    ex s2 be Element of S st s1 = s2 & s2 |= f by A4;
    then s |= g by A5,Th28;
    hence thesis by A2;
  end;
  for x being object holds x in SIGMA(g) implies x in Pred(H,R)
  proof
    let x be object;
    assume x in SIGMA(g);
    then consider s be Element of S such that
A6: x=s and
A7: s|= g;
    consider s1 being Element of S such that
A8: [s,s1] in R and
A9: s1 |= f by A7,Th28;
    s1 in H by A9;
    hence thesis by A6,A8;
  end;
  hence thesis by A1,TARSKI:2;
end;
