theorem Th39:
  rng chi(A,X) c= {0,1}
proof
  let y be object;
  assume y in rng chi(A,X);
  then consider x being object such that
A1: x in dom chi(A,X) and
A2: y = chi(A,X).x by FUNCT_1:def 3;
A3: x in A or not x in A;
  x in X by A1,Def3;
  then y = {} or y = 1 by A2,A3,Def3;
  hence thesis by TARSKI:def 2;
end;
