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 Th28:
  a <= 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 & c in ['a,b'] & d in ['a,b']
  implies integral(f-g,c,d) = integral(f,c,d) - integral(g,c,d)
  proof
    assume A1: a<=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 & c in ['a,b'] & d in ['a,b'];
A2: -g is_integrable_on ['a,b'] & (-g) | ['a,b'] is bounded by A1,Th12;
A3: dom g = dom -g by NFCONT_4:def 3;
    f-g = f+-g by Lm1;
    hence integral(f-g,c,d) = integral(f,c,d) + integral(-g,c,d)
    by A1,A2,A3,Th27
    .= integral(f,c,d) - integral(g,c,d) by A1,Th26;
   end;
