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 Th45:
  for a,b,c,d,e be Real,
              f be PartFunc of REAL,REAL-NS n
    st (a<=b & c <= d & 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'] &
     for x be Real st x in ['c,d'] holds ||. f/.x .|| <= e)
  holds
    ||. integral(f,c,d) .|| <= e * (d-c) &
    ||. integral(f,d,c).|| <= e * (d-c)
proof
  let a,b,c,d,e be Real,
  f be PartFunc of REAL,REAL-NS n;
  assume
A1: a<=b & c <= d & 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']
  & for x be Real st x in ['c,d'] holds ||. f/.x .|| <= e;
A2: ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3; then
A3: ex g be Real st c = g & a<=g & g <= b by A1;
A4: ex g be Real st d=g & a<=g & g <= b by A2,A1;
  reconsider f1=f as PartFunc of REAL,REAL n by REAL_NS1:def 4;
A5: f1 | ['a,b'] is bounded by A1,INTEGR19:34;
A6: f1 is_integrable_on ['a,b'] by A5,A1,INTEGR19:43;
A7: |. f1 .| is_integrable_on ['a,b'] by A1,NFCONT_4:9;
  now let x be Real;
    assume A8: x in ['c,d']; then
  A9: ||. f/.x .|| <= e by A1;
  A10: ['c,d'] c= dom f by A1,INTEGR19:2,A3,A4; then
    f/.x = f.x by A8,PARTFUN1:def 6
        .= f1/.x by A8,A10,PARTFUN1:def 6;
    hence |. f1/.x .| <= e by A9,REAL_NS1:1;
  end; then
  |. integral(f1,c,d) .| <= e * (d-c) & |. integral(f1,d,c).| <= e * (d-c)
    by A1,A5,A6,A7,INTEGR19:24;
  hence
  ||. integral(f,c,d) .|| <= e * (d-c) & ||. integral(f,d,c).|| <= e * (d-c)
    by REAL_NS1:1,A1,INTEGR19:48;
end;
