reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem Th21:
  for n be non zero Element of NAT, h be PartFunc of REAL,REAL n
  st a<=b & ['a,b'] c= dom h & h is_integrable_on ['a,b']
  & |.h.| is_integrable_on ['a,b'] & h| ['a ,b'] is bounded
  holds |. integral(h,a,b).| <= integral(|.h.|,a,b)
  proof
    let n be non zero Element of NAT, h be PartFunc of REAL,REAL n;
    assume A1: a<=b & ['a,b'] c= dom h & h is_integrable_on ['a,b']
    & |.h.| is_integrable_on ['a,b'] & h| ['a ,b'] is bounded;
    ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3; then
A2: integral(h,a,b) = integral(h,['a,b']) by INTEGR15:19;
    integral(|.h.|,a,b) = integral (|.h.|,['a,b']) by A1,INTEGRA5:def 4;
    hence thesis by Th20,A1,A2;
  end;
