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 Th13:
  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
  implies
  f - g is_integrable_on ['c,d'] & (f - g) | ['c,d'] is bounded
  proof
    assume that
A1: a <= c & c <= d & d <= b and
A2: f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] &
    f| ['a,b'] is bounded & g| ['a,b'] is bounded and
A3: ['a,b'] c= dom f & ['a,b'] c= dom g;
A4: dom g = dom -g by NFCONT_4:def 3;
A5: f-g = f+-g by Lm1;
    a <= d by A1,XXREAL_0:2; then
    a <= b by A1,XXREAL_0:2;
    then -g is_integrable_on ['a,b'] & (-g) | ['a,b'] is bounded by A2,A3,Th12;
    hence thesis by A5,A1,A2,A3,A4,Th10;
  end;
