reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;
reserve X,Y for set;

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;
