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 Th20:
  for n be non zero Element of NAT, h be PartFunc of REAL,REAL n
  st A c= dom h & h|A is bounded &
  h is_integrable_on A & |. h .| is_integrable_on A
  holds |.integral(h,A) .| <= integral(|. h .|,A)
  proof
    let n be non zero Element of NAT, h be PartFunc of REAL,REAL n;
    assume A1: A c= dom h & h|A is bounded
    & h is_integrable_on A & |. h .| is_integrable_on A;
    set f = |. h .|;
    reconsider hA = h|A as Function of A,REAL n by A1,Lm9;
A2: integral(h,A) = integral(hA) by INTEGR15:14;
    A c= dom f by A1,NFCONT_4:def 2; then
    reconsider fA = f|A as Function of A,REAL by Lm5;
A3: fA is integrable by A1;
A4: hA is integrable by A1,INTEGR15:13;
    |. hA .| = |. (h|A) .| by Th17
    .= f|A by Th18,A1;
    hence thesis by A2,A3,A4,A1,Th16,Lm8;
  end;
