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
  for f,g be PartFunc of REAL,REAL
  st a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] &
  g is_integrable_on ['a,b'] & f| ['a,b'] is bounded &
  g| ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g
  holds f-g is_integrable_on ['c,d'] & (f-g) | ['c,d'] is bounded
  proof
    let f,g be PartFunc of REAL,REAL;
    assume A1: a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] &
    g is_integrable_on ['a,b'] & f| ['a,b'] is bounded &
    g| ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g;
A2: ((-1)(#)g) | ['a,b'] is bounded by A1,RFUNCT_1:80;
A3: ['a,b'] c= dom ((-1)(#)g) by A1,VALUED_1:def 5;
    (-1)(#)g is_integrable_on ['a,b'] by A1,INTEGRA6:9;
    hence thesis by A1,A2,A3,INTEGRA6:19;
  end;
