
theorem Th10:
  for X being non empty set,
      f being Function of X,COMPLEX st f | X is bounded holds
                    PreNorms f is bounded_above
proof
  let X be non empty set,
      f be Function of X,COMPLEX;
A1: dom |.f.| = dom f by VALUED_1:def 11;
A2: dom (|.f.| | X) = X /\ (dom |.f.|) by RELAT_1:61;
A3: |.f.| | X = |.(f | X).| by RFUNCT_1:46;
  assume f|X is bounded;
  then |.f.| | X is bounded by A3,Lm2;
  then consider p being Real such that
A4: for c being object st c in dom (|.f.| | X) holds
               |.((|.f.| | X).c).| <= p by RFUNCT_1:72;
A5:now
   let c be Element of X;
   assume
A6:c in X /\ (dom f);
   |.((|.f.| | X).c).| = |.(|.f.|.c).| by A1,A2,A6,FUNCT_1:47
                        .= |.|.(f.c).|.| by VALUED_1:18;
   hence |.(f.c).| <= p by A1,A2,A4,A6;
  end;
A7:X /\ dom f = X /\ X by FUNCT_2:def 1;
A8:now let r be ExtReal;
   assume r in PreNorms f; then
   consider t be Element of X such that
A9:r=|.f.t.|;
   thus r <=p by A7,A9,A5;
  end;
  p is UpperBound of (PreNorms f) by A8,XXREAL_2:def 1;
  hence thesis by XXREAL_2:def 10;
end;
