reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

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