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 Th6:
  for f be PartFunc of REAL,REAL holds
  a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] &
  f | [' a,b '] is bounded & ['a,b'] c= dom f
  implies r(#)f is_integrable_on ['c,d'] & (r(#)f) | ['c,d'] is bounded
  proof
    let f be PartFunc of REAL,REAL;
    assume that
A1: a <= c and
A2: c <= d & d <= b and
A3: f is_integrable_on ['a,b'] & f | ['a,b'] is bounded and
A4: ['a,b'] c= dom f;
A5: f | ['c,d'] is bounded by A1,A2,A3,A4,INTEGRA6:18;
A6: ['c,d'] c= dom f by A2,A1,Th2,A4;
    f is_integrable_on ['c,d'] by A1,A2,A3,A4,INTEGRA6:18;
    hence r(#)f is_integrable_on ['c,d'] by A5,A6,INTEGRA6:9;
    thus thesis by A5,RFUNCT_1:80;
  end;
