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 Th50:
  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'] &
  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']
  holds integral(f-g,c,d) = integral(f,c,d) - integral(g,c,d)
  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']
    & 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'];
    reconsider f1=f, g1=g as PartFunc of REAL,REAL n by REAL_NS1:def 4;
A2: f1| ['a,b'] is bounded by Th34,A1;
A3: g1| ['a,b'] is bounded by Th34,A1;
A4: f1 is_integrable_on ['a,b'] by Th43,A2,A1;
A5: g1 is_integrable_on ['a,b'] by Th43,A3,A1;
A6: f1-g1 = f-g by NFCONT_4:10;
A7: integral(f1,c,d) = integral(f,c,d) &
    integral(g1,c,d) = integral(g,c,d) by A1,Th48;
A8: ['a,b'] c= dom(f-g) by A1,A6,Th5;
     f1-g1 is_integrable_on ['a,b'] & (f1-g1)| ['a,b'] is bounded
     by A1,A2,A3,A4,A5,Th13;
     then f-g is_integrable_on ['a,b'] & (f-g)| ['a,b'] is bounded
     by Th43,A6,A8,Th34;
     hence integral(f-g,c,d) = integral(f1-g1,c,d) by A1,A8,Th48,NFCONT_4:10
     .= integral(f1,c,d) - integral(g1,c,d) by A1,A2,A3,A4,A5,Th28
     .= integral(f,c,d) - integral(g,c,d) by A7,REAL_NS1:5;
  end;
