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 Th33:
  for f,g being Assign of BASSModel(R,BASSIGN) holds SIGMA('not' f)
  = S \ SIGMA(f) & SIGMA(f '&' g) = SIGMA(f) /\ SIGMA(g) & SIGMA(f 'or' g) =
  SIGMA(f) \/ SIGMA(g)
proof
  let f,g be Assign of BASSModel(R,BASSIGN);
A1: for s being object holds s in SIGMA('not' f) iff s in S \ SIGMA(f)
  proof
    let s be object;
A2: s in SIGMA('not' f) implies s in S \ SIGMA(f)
    proof
      assume
A3:   s in SIGMA('not' f);
      then
A4:   ex x being Element of S st x = s & x |= 'not' f;
      reconsider s as Element of S by A3;
      not s in SIGMA(f)
      proof
        assume s in SIGMA(f);
        then ex y being Element of S st y = s & y |= f;
        hence contradiction by A4,Th12;
      end;
      hence thesis by XBOOLE_0:def 5;
    end;
    s in S \ SIGMA(f) implies s in SIGMA('not' f)
    proof
      assume
A5:   s in S \ SIGMA(f);
      then reconsider s as Element of S;
      not s in SIGMA(f) by A5,XBOOLE_0:def 5;
      then s |/= f;
      then s |= 'not' f by Th12;
      hence thesis;
    end;
    hence thesis by A2;
  end;
A6: for s being object
    holds s in SIGMA(f 'or' g) iff s in SIGMA(f) \/ SIGMA(g )
  proof
    let s be object;
A7: s in SIGMA(f) \/ SIGMA(g) implies s in SIGMA(f 'or' g)
    proof
      assume
A8:   s in SIGMA(f) \/ SIGMA(g);
      per cases by A8,XBOOLE_0:def 3;
      suppose
A9:     s in SIGMA(f);
        then
A10:    ex x being Element of S st x = s & x |= f;
        reconsider s as Element of S by A9;
        s |= f 'or' g by A10,Th17;
        hence thesis;
      end;
      suppose
A11:    s in SIGMA(g);
        then
A12:    ex x being Element of S st x = s & x |= g;
        reconsider s as Element of S by A11;
        s |= f 'or' g by A12,Th17;
        hence thesis;
      end;
    end;
    s in SIGMA(f 'or' g) implies s in SIGMA(f) \/ SIGMA(g)
    proof
      assume
A13:  s in SIGMA(f 'or' g);
      then
A14:  ex x being Element of S st x = s & x |= f 'or' g;
      reconsider s as Element of S by A13;
      per cases by A14,Th17;
      suppose
        s|= f;
        then s in SIGMA(f);
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        s|= g;
        then s in SIGMA(g);
        hence thesis by XBOOLE_0:def 3;
      end;
    end;
    hence thesis by A7;
  end;
  for s being object holds s in SIGMA(f '&' g) iff s in SIGMA(f) /\ SIGMA(g)
  proof
    let s be object;
    thus s in SIGMA(f '&' g) implies s in SIGMA(f) /\ SIGMA(g)
    proof
      assume
A15:  s in SIGMA(f '&' g);
      then
A16:  ex x being Element of S st x = s & x |= f '&' g;
      reconsider s as Element of S by A15;
      s|= g by A16,Th13;
      then
A17:  s in SIGMA(g);
      s|= f by A16,Th13;
      then s in SIGMA(f);
      hence thesis by A17,XBOOLE_0:def 4;
    end;
    assume
A18: s in SIGMA(f) /\ SIGMA(g);
    then
A19: s in SIGMA(g) by XBOOLE_0:def 4;
    s in SIGMA(f) by A18,XBOOLE_0:def 4;
    then
A20: ex x being Element of S st x = s & x |= f;
    reconsider s as Element of S by A18;
    ex y being Element of S st y = s & y |= g by A19;
    then s |= f '&' g by A20,Th13;
    hence thesis;
  end;
  hence thesis by A1,A6,TARSKI:2;
end;
