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 Th37:
  for f be Function of A,REAL n, g be Function of A,REAL-NS n,
  D be Division of A,
  p be middle_volume of f,D,
  q be middle_volume of g,D
  st f=g & p=q
  holds middle_sum(f,p) = middle_sum(g,q)
  proof
    let f be Function of A,REAL n,
    g be Function of A,REAL-NS n,
    D be Division of A,
    p be middle_volume of f,D,
    q be middle_volume of g,D;
    assume A1: f=g & p=q;
    for i be Element of NAT st i in Seg n holds
    ex Pi be FinSequence of REAL st Pi=proj(i,n)*p
    & (middle_sum(f,p)).i = Sum(Pi) by INTEGR15:def 6;
    hence middle_sum(f,p) = middle_sum(g,q) by Lm16,A1;
  end;
