reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;
reserve
  Z for open Subset of REAL,
 y0 for VECTOR of REAL-NS n,
  G for Function of REAL-NS n,REAL-NS n;

theorem Th44:
  for a,b,c,d be Real,
      f be PartFunc of REAL,REAL-NS n
    st a <= b & f is_integrable_on ['a,b']
   & ||. f .|| is_integrable_on ['a,b']
   & f| ['a,b'] is bounded & ['a,b'] c= dom f
   & c in ['a,b'] & d in ['a,b']
  holds ||. f .|| is_integrable_on ['min(c,d),max(c,d)'] &
    (||. f .||) | ['min(c,d),max(c,d)'] is bounded &
    ||. integral(f,c,d) .|| <= integral((||. f .||),min(c,d),max(c,d))
proof
  let a,b,c,d be Real,
      f be PartFunc of REAL,REAL-NS n;
  assume A1: a <= b & f is_integrable_on ['a,b']
           & ||. f .|| is_integrable_on ['a,b']
           & f| ['a,b'] is bounded & ['a,b'] c= dom f
           & c in ['a,b'] & d in ['a,b'];
  reconsider f1=f as PartFunc of REAL,REAL n by REAL_NS1:def 4;
A2: f1 | ['a,b'] is bounded by A1,INTEGR19:34;
A3: f1 is_integrable_on ['a,b'] by A2,A1,INTEGR19:43;
A4: ||. f .|| = |. f1 .| by NFCONT_4:9;
A5: |. f1 .| is_integrable_on ['a,b'] by A1,NFCONT_4:9;
  |. f1 .| is_integrable_on ['min(c,d),max(c,d)'] &
  (|. f1 .|) | ['min(c,d),max(c,d)'] is bounded &
  |. integral(f1,c,d) .| <= integral((|. f1 .|),min(c,d),max(c,d))
  by A1,A2,A3,A5,INTEGR19:22;
  hence thesis by A4,REAL_NS1:1,A1,INTEGR19:48;
end;
