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 Th1920:
  for f be PartFunc of REAL,the carrier of Y
    st A c= dom f & f|A is bounded & f is_integrable_on A
     & ||.f.|| is_integrable_on A
  holds ||.integral(f,A).|| <= integral(||.f.||,A)
proof
   let f be PartFunc of REAL,the carrier of Y;
   assume A1: A c= dom f & f|A is bounded & f is_integrable_on A
    & ||.f.|| is_integrable_on A;
   set g = ||.f.||;
   reconsider fA = f|A as Function of A,the carrier of Y by A1;
A2:integral(f,A) = integral fA by A1,INTEGR18:9;
   A c= dom g by A1,NORMSP_0:def 2; then
   reconsider gA = g|A as Function of A,REAL by Lm3;
A3:gA is integrable by A1;
   fA is bounded & ||. fA .|| = g|A by Th1918,A1;
   hence thesis by A2,A3,A1,Lm8;
end;
