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 Th36:
  for f be Function of A,REAL n, g be Function of A,REAL-NS n,
      D be Division of A, p be FinSequence of REAL n,
      q be FinSequence of REAL-NS n st f=g & p=q
  holds p is middle_volume of f,D iff q is middle_volume of g,D
  proof
    let f be Function of A,REAL n,
    g be Function of A,REAL-NS n,
    D be Division of A,
    p be FinSequence of REAL n,
    q be FinSequence of REAL-NS n;
    assume A1: f=g & p=q;
    thus p is middle_volume of f,D
    implies q is middle_volume of g,D
    proof
      assume A2: p is middle_volume of f,D; then
A3:   len q = len D by A1,INTEGR15:def 5;
      for i be Nat st i in dom D holds
      ex c be Point of REAL-NS n st c in rng (g|divset(D,i)) &
      q.i= (vol divset(D,i)) * c
      proof
        let i be Nat;
        assume i in dom D;
        then consider r be Element of REAL n such that
A4:     r in rng (f|divset(D,i)) & p.i=vol divset(D,i)*r
        by A2,INTEGR15:def 5;
        reconsider c = r as Point of REAL-NS n by REAL_NS1:def 4;
        take c;
        thus thesis by A4,A1,REAL_NS1:3;
      end;
      hence q is middle_volume of g,D by A3,INTEGR18:def 1;
    end;
    thus q is middle_volume of g,D implies p is middle_volume of f,D
    proof
      assume A5: q is middle_volume of g,D; then
A6:   len p = len D by A1,INTEGR18:def 1;
      for i be Nat st i in dom D holds ex r be Element of REAL n st r in
      rng (f|divset(D,i)) & p.i=vol divset(D,i)*r
      proof
        let i be Nat;
        assume A7: i in dom D;
        consider c be Point of REAL-NS n such that
A8:     c in rng (g|divset(D,i)) & q.i=vol divset(D,i)*c
        by A5,A7,INTEGR18:def 1;
        reconsider r = c as Element of REAL n by REAL_NS1:def 4;
        take r;
        thus thesis by A8,A1,REAL_NS1:3;
      end;
      hence p is middle_volume of f,D by A6,INTEGR15:def 5;
    end;
  end;
