
theorem Th4:
for X be non empty set, A be set, r be Real holds
  rng chi(r,A,X) c= {0,r} & chi(r,A,X) is without+infty without-infty
proof
   let X be non empty set, A be set, r be Real;
   now let y be object;
    assume y in rng chi(r,A,X); then
    consider x be object such that
A1:  x in dom chi(r,A,X) & y = chi(r,A,X).x by FUNCT_1:def 3;
    per cases;
    suppose x in A; then
     chi(r,A,X).x = r by A1,Def1;
     hence y in {0,r} by A1,TARSKI:def 2;
    end;
    suppose not x in A; then
     chi(r,A,X).x = 0 by A1,Def1;
     hence y in {0,r} by A1,TARSKI:def 2;
    end;
   end;
   hence rng chi(r,A,X) c= {0,r};
   chi(A,X) is without+infty without-infty by Th3; then
   r(#)chi(A,X) is without+infty without-infty;
   hence chi(r,A,X) is without+infty without-infty by Th1;
end;
