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 Th43:
  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
  holds f is_integrable_on A iff g is_integrable_on 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;
    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: h is bounded by A1;
    hereby assume f is_integrable_on A;
      then h is integrable by INTEGR15:13;
      then k is integrable by A2,Th41;
      hence g is_integrable_on A by A1;
    end;
    assume g is_integrable_on A;
    then k is integrable by A1;
    then h is integrable by A2,Th41;
    hence thesis by INTEGR15:13;
  end;
