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 Th39:
  for f be Function of A,REAL n,
  g be Function of A,REAL-NS n,
  T be DivSequence of A,
  S be middle_volume_Sequence of f,T,
  U be middle_volume_Sequence of g,T
  st f=g & S=U
  holds middle_sum(f,S) = middle_sum(g,U)
  proof
    let f be Function of A,REAL n,
    g be Function of A,REAL-NS n,
    T be DivSequence of A,
    S be middle_volume_Sequence of f,T,
    U be middle_volume_Sequence of g,T;
    assume A1:f=g & S=U;
A2: dom (middle_sum(f,S)) = NAT by FUNCT_2:def 1
    .=dom (middle_sum(g,U)) by FUNCT_2:def 1;
    now let x be object;
      assume x in dom (middle_sum(f,S));
      then reconsider n=x as Element of NAT;
A3:   middle_sum(f,S.n) = middle_sum(g,U.n) by A1,Th37;
      thus (middle_sum(f,S)).x = middle_sum(f,S.n) by INTEGR15:def 8
      .=(middle_sum(g,U)).x by A3,INTEGR18:def 4;
    end;
    hence thesis by A2,FUNCT_1:2;
  end;
