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 Th53:
  for f be PartFunc of REAL,REAL-NS n
  st a<=b & f is_integrable_on ['a,b'] & f| ['a,b'] is bounded &
  ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b']
  holds integral(f,a,d) = integral(f,a,c) + integral(f,c,d)
  proof
    let f be PartFunc of REAL,REAL-NS n;
    assume
A1: a<=b & f is_integrable_on ['a,b'] & f| ['a,b'] is bounded &
    ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'];
    ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3;
    then
A2: a in ['a,b'] & b in ['a,b'] by A1;
    reconsider f1=f as PartFunc of REAL,REAL n by REAL_NS1:def 4;
A3: f1 | ['a,b'] is bounded by A1,Th34;
A4: f1 is_integrable_on ['a,b'] by Th43,A3,A1;
A5: integral(f1,a,d) = integral(f1,a,c) + integral(f1,c,d) by A3,A4,A1,Th31;
A6: integral(f,a,d)= integral(f1,a,d) by A2,Th48,A1;
A7: integral(f,a,c)= integral(f1,a,c) by A2,Th48,A1;
    integral(f,c,d)= integral(f1,c,d) by Th48,A1;
    hence integral(f,a,d) = integral(f,a,c) + integral(f,c,d)
    by A7,A5,A6,REAL_NS1:2;
  end;
