
theorem
  for X be non empty set, r be non zero Real, f be Function of X,ExtREAL
    holds f is without-infty without+infty iff
      r(#)f is without-infty without+infty
proof
   let X be non empty set, r be non zero Real, f be Function of X,ExtREAL;
   per cases;
   suppose r > 0;
    hence f is without-infty without+infty
     iff r(#)f is without-infty without+infty by Th16,Th18;
   end;
   suppose r < 0;
    hence f is without-infty without+infty
     iff r(#)f is without-infty without+infty by Th17,Th19;
   end;
end;
