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 Th44:
  for f be PartFunc of REAL,REAL n, g be PartFunc of REAL,REAL-NS n
  st f=g & f|A is bounded & A c= dom f & f is_integrable_on A
  holds g is_integrable_on A & integral(f,A) = integral(g,A)
  proof
    let f be PartFunc of REAL,REAL n,
    g be PartFunc of REAL,REAL-NS n;
    assume A1: f=g & f|A is bounded & A c= dom f
    & f is_integrable_on A;
    hence g is_integrable_on A by Th43;
    reconsider h= f|A as Function of A,REAL n by Lm3,A1;
    reconsider k=h as Function of A,REAL-NS n by REAL_NS1:def 4;
A2: integral(f,A) = integral(h) by INTEGR15:14;
A3: h is bounded by A1;
    h is integrable by A1,INTEGR15:13; then
    integral(h) = integral(k) by A3,Th42;
    hence thesis by A2,A1,INTEGR18:9;
  end;
