
theorem
  for X be non empty set, f be Function of X,ExtREAL holds
    0(#)f = X --> 0
    & 0(#)f is without-infty without+infty
proof
   let X be non empty set, f be Function of X,ExtREAL;
A1:X --> 0 is Function of X,ExtREAL by FUNCOP_1:45,XXREAL_0:def 1;
   for x be Element of X holds (0(#)f).x = (X --> 0).x
   proof
    let x be Element of X;
    x in X; then
    x in dom(0(#)f) by FUNCT_2:def 1; then
    (0(#)f).x = 0 * f.x by MESFUNC1:def 6;
    hence (0(#)f).x = (X --> 0).x by FUNCOP_1:7;
   end;
   hence 0(#)f = X --> 0 by A1,FUNCT_2:def 8;
   hence 0(#)f is without-infty without+infty by Th21;
end;
