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 Th40:
  for f be Function of A,REAL n, g be Function of A,REAL-NS n,
  I be Element of REAL n, J be Point of REAL-NS n
  st f=g & I = J holds
  (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)
  iff
  (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))=J)
  proof
    let f be Function of A,REAL n,
    g be Function of A,REAL-NS n,
    I be Element of REAL n,
    J be Point of REAL-NS n;
    assume A1: f=g & I = J;
    hereby assume 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;
A3:   the carrier of REAL-NS n = REAL n by REAL_NS1:def 4;
      thus 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))=J
      proof let T be DivSequence of A, S0 be middle_volume_Sequence of g,T;
        assume A4:delta(T) is convergent & lim delta(T)=0;
        reconsider S=S0 as middle_volume_Sequence of f,T by A3,A1,Th38;
        middle_sum(f,S) is convergent & lim (middle_sum(f,S))=I by A2,A4;
        hence thesis by A1,Th39;
      end;
    end;
    assume
A5: 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))=J;
A6: the carrier of REAL-NS n = REAL n by REAL_NS1:def 4;
    thus 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))=I
    proof
      let T be DivSequence of A, S0 be middle_volume_Sequence of f,T;
      assume A7:delta(T) is convergent & lim delta(T)=0;
      reconsider S=S0  as middle_volume_Sequence of g,T by A6,A1,Th38;
      middle_sum(g,S) is convergent & lim (middle_sum(g,S))=I by A1,A5,A7;
      hence thesis by A1,Th39;
    end;
  end;
