theorem
  f|Y is constant implies (for z holds z(#)f is_bounded_on Y) & -f
  is_bounded_on Y & ||.f.|||Y is bounded
proof
  assume
A1: f|Y is constant;
  hereby
    let z;
    (z(#)f)|Y is constant by A1,Th52;
    hence z(#)f is_bounded_on Y by Th54;
  end;
  (-f)|Y is constant by A1,Th53;
  hence -f is_bounded_on Y by Th54;
  ||.f.|||Y is constant by A1,Th53;
  hence thesis;
end;
