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 Th38:
  for f be Function of A,REAL n,
  g be Function of A,REAL-NS n,
  T be DivSequence of A,
  p be sequence of (REAL n)*,
  q be sequence of (the carrier of REAL-NS n)*
  st f=g & p=q
  holds p is middle_volume_Sequence of f,T
  iff  q is middle_volume_Sequence of g,T
  proof
    let f be Function of A,REAL n,
    g be Function of A,REAL-NS n,
    T be DivSequence of A,
    p be sequence of (REAL n)*,
    q be sequence of (the carrier of REAL-NS n)*;
    assume A1:f=g & p=q;
    hereby assume A2: p is middle_volume_Sequence of f,T;
      for k be Element of NAT holds q.k is middle_volume of g,T.k
      proof
        let k be Element of NAT;
A3:     p.k is middle_volume of f,T.k by A2,INTEGR15:def 7;
        thus q.k is middle_volume of g,T.k by A1,A3,Th36;
      end;
      hence q is middle_volume_Sequence of g,T by INTEGR18:def 3;
    end;
    assume A4: q is middle_volume_Sequence of g,T;
    for k be Element of NAT holds p.k is middle_volume of f,T.k
    proof
      let k be Element of NAT;
A5:   q.k is middle_volume of g,T.k by A4,INTEGR18:def 3;
      thus p.k is middle_volume of f,T.k by A1,A5,Th36;
    end;
    hence p is middle_volume_Sequence of f,T by INTEGR15:def 7;
  end;
