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;
