theorem Th32:
  for T being Subset of S holds SIGMA(Tau(T,R,BASSIGN)) = T
proof
  let T be Subset of S;
  set f = Tau(T,R,BASSIGN);
  set U = SIGMA(f);
A1: U = { s where s is Element of S : (Fid(f,S)).s=TRUE } by Lm40;
  for s being object holds s in U iff s in T
  proof
    let s be object;
    thus s in U implies s in T
    proof
      assume s in U;
      then ex t being Element of S st s = t & (Fid(f,S)).t=TRUE by A1;
      then chi(T,S).s =TRUE by Def64;
      hence thesis by FUNCT_3:36;
    end;
    assume
A2: s in T;
    then (Fid(f,S)).s = chi(T,S).s by Def64
      .= 1 by A2,FUNCT_3:def 3;
    hence thesis by A1,A2;
  end;
  hence thesis by TARSKI:2;
end;
