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 Th45:
  for f be PartFunc of REAL,REAL n, g be PartFunc of REAL,REAL-NS n
  st f=g & a <= b & f| ['a,b'] is bounded & ['a,b'] c= dom f
  & f is_integrable_on ['a,b']
  holds integral(f,a,b) = integral(g,a,b)
  proof
    let f be PartFunc of REAL,REAL n, g be PartFunc of REAL,REAL-NS n;
    assume A1:f=g & a <= b & f | ['a,b'] is bounded & ['a,b'] c= dom f
    & f is_integrable_on ['a,b'];
A2: ['a,b'] =[.a,b.] by A1,INTEGRA5:def 3;
A3: integral(g,['a,b'])= integral(f,['a,b']) by A1,Th44;
    integral(g,['a,b']) = integral(g,a,b) by A2,INTEGR18:16;
    hence thesis by A3,A2,INTEGR15:19;
  end;
