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 Th41:
  for f be Function of A,REAL n, g be Function of A,REAL-NS n
  st f=g & f is bounded
  holds f is integrable iff g is integrable
  proof
    let f be Function of A,REAL n,
    g be Function of A,REAL-NS n;
    assume A1: f=g & f is bounded;
    hereby assume f is integrable;
      then consider I be Element of REAL n such that
A2:   for T be DivSequence of A, S be middle_volume_Sequence of f,T
      st delta(T) is convergent & lim delta(T)=0 holds
      middle_sum(f,S) is convergent & lim (middle_sum(f,S))=I
      by A1,INTEGR15:12;
      reconsider I0 = I as Point of REAL-NS n by REAL_NS1:def 4;
      I=I0;
      then for T be DivSequence of A, S0 be middle_volume_Sequence of g,T
      st delta(T) is convergent & lim delta(T)=0 holds
      middle_sum(g,S0) is convergent & lim (middle_sum(g,S0))=I0 by A2,A1,Th40;
      hence g is integrable;
    end;
    assume g is integrable;
    then consider I be  Point of REAL-NS n such that
A3: for T be DivSequence of A, S be middle_volume_Sequence of g,T
    st delta(T) is convergent & lim delta(T)=0 holds
    middle_sum(g,S) is convergent & lim (middle_sum(g,S))=I;
    reconsider I0 = I as Element of  REAL n by REAL_NS1:def 4;
    I0=I;
    then
    for T be DivSequence of A, S0 be middle_volume_Sequence of f,T
    st delta(T) is convergent & lim delta(T)=0
    holds middle_sum(f,S0) is convergent
    & lim (middle_sum(f,S0))=I0 by A3,A1,Th40;
    hence f is integrable by A1,INTEGR15:12;
  end;
