reserve Z for RealNormSpace;
reserve a,b,c,d,e,r for Real;
reserve A,B for non empty closed_interval Subset of REAL;
reserve X,Y for RealBanachSpace;
reserve E for Point of Y;

theorem Th4:
  for Y be RealNormSpace,
      f be continuous PartFunc of REAL,the carrier of Y
     st a <= b & ['a,b'] c= dom f
      holds ||.f.|| is_integrable_on ['a,b']
proof
   let Y be RealNormSpace,
       f be continuous PartFunc of REAL,the carrier of Y;
   assume A1: a <= b & ['a,b'] c= dom f;
P11: f | [' a,b '] is continuous;
   ['a,b'] c= dom ||.f.|| by NORMSP_0:def 3,A1;
   hence ||.f.|| is_integrable_on ['a,b'] by P11,A1,NFCONT_3:22,INTEGRA5:11;
end;
