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 Th1921:
  for f be PartFunc of REAL,the carrier of Y
    st a<=b & ['a,b'] c= dom f & f is_integrable_on ['a,b']
  & ||.f.|| is_integrable_on ['a,b'] & f| ['a ,b'] is bounded
  holds ||.integral(f,a,b).|| <= integral(||.f.||,a,b)
proof
   let f be PartFunc of REAL,the carrier of Y;
   assume A1: a<=b & ['a,b'] c= dom f & f is_integrable_on ['a,b']
     & ||.f.|| is_integrable_on ['a,b'] & f| ['a ,b'] is bounded;
   ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3; then
A3:integral(f,a,b) = integral(f,['a,b']) by INTEGR18:16;
   integral(||.f.||,a,b) = integral (||.f.||,['a,b']) by A1,INTEGRA5:def 4;
   hence thesis by Th1920,A1,A3;
end;
