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-NS n
  st a <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b']
  & ['a,b'] c= dom f & ['a,b'] c= dom g
  holds integral(f+g,a,b) = integral(f,a,b) + integral(g,a,b) &
  integral(f-g,a,b) = integral(f,a,b) - integral(g,a,b)
  proof
    let f,g be PartFunc of REAL,REAL-NS n;
    assume
A1: a <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b']
    & ['a,b'] c= dom f & ['a,b'] c= dom g;
A2: f+g is_integrable_on ['a,b'] & integral(f + g,['a,b'])
    = integral(f,['a,b']) + integral(g,['a,b']) by A1,INTEGR18:14;
A3: f-g is_integrable_on ['a,b'] & integral(f - g,['a,b'])
    = integral(f,['a,b']) - integral(g,['a,b']) by A1,INTEGR18:15;
A4: ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3;
    thus integral(f + g,a,b)
    = integral(f + g,['a,b']) by A4,INTEGR18:16
    .= integral(f,a,b) + integral(g,['a,b']) by A4,A2,INTEGR18:16
    .= integral(f,a,b) + integral(g,a,b) by A4,INTEGR18:16;
    thus integral(f - g,a,b)
    = integral(f - g,['a,b']) by A4,INTEGR18:16
    .= integral(f,a,b) - integral(g,['a,b']) by A4,A3,INTEGR18:16
    .= integral(f,a,b) - integral(g,a,b) by A4,INTEGR18:16;
  end;
