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 T1,T2 being Subset of S holds Tau(T1,R,BASSIGN) = Tau(T2,R,BASSIGN
  ) implies T1=T2
proof
  let T1,T2 be Subset of S;
  set h1 = Tau(T1,R,BASSIGN);
  set h2 = Tau(T2,R,BASSIGN);
  assume
A1: h1 = h2;
A2: for s being object holds s in T2 implies s in T1
  proof
    let s be object;
    assume
A3: s in T2;
    then chi(T2,S).s =1 by FUNCT_3:def 3;
    then (Fid(h2,S)).s =TRUE by A3,Def64;
    then chi(T1,S).s = 1 by A1,A3,Def64;
    hence thesis by FUNCT_3:36;
  end;
  for s being object holds s in T1 implies s in T2
  proof
    let s be object;
    assume
A4: s in T1;
    then chi(T1,S).s =1 by FUNCT_3:def 3;
    then (Fid(h1,S)).s =TRUE by A4,Def64;
    then chi(T2,S).s = 1 by A1,A4,Def64;
    hence thesis by FUNCT_3:36;
  end;
  hence thesis by A2,TARSKI:2;
end;
