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 Th31:
  for f being Assign of BASSModel(R,BASSIGN) holds Tau(SIGMA(f),R, BASSIGN) = f
proof
  let f be Assign of BASSModel(R,BASSIGN);
  set T = SIGMA(f);
  set g = Tau(T,R,BASSIGN);
A1: T = { s where s is Element of S : (Fid(f,S)).s=TRUE } by Lm40;
  for s being object st s in S holds Fid(f,S).s= Fid(g,S).s
  proof
    let s be object;
    assume s in S;
    then reconsider s as Element of S;
    per cases;
    suppose
A2:   s in T;
A3:   Fid(g,S).s = chi(T,S).s by Def64
        .= 1 by A2,FUNCT_3:def 3;
      ex x being Element of S st x=s & (Fid(f,S)).x=TRUE by A1,A2;
      hence thesis by A3;
    end;
    suppose
A4:   not s in T;
A5:   (Fid(f,S)).s=FALSE
      proof
        assume (Fid(f,S)).s <> FALSE;
        then (Fid(f,S)).s= TRUE by TARSKI:def 2;
        hence contradiction by A1,A4;
      end;
      Fid(g,S).s = chi(T,S).s by Def64
        .=0 by A4,FUNCT_3:def 3;
      hence thesis by A5;
    end;
  end;
  hence thesis by Lm42,FUNCT_2:12;
end;
