
theorem LM84:
  for A be non empty closed_interval Subset of REAL,
      D be Division of A,
      m be Function of A, BoundedFunctions(A),
      rho be Function of A,REAL
    ex s be FinSequence of
          R_Normed_Algebra_of_BoundedFunctions
            the carrier of ClstoCmp(A)
      st len s = len D
       & for i be Nat st i in dom s holds
           s.i = sgn(Dp2(rho,D,i+1) - Dp2(rho,D,i))
                * (Dp1(m,D,i+1) - Dp1(m,D,i))
proof
  let A be non empty closed_interval Subset of REAL,
      D be Division of A,
      m be Function of A, BoundedFunctions(A),
      rho be Function of A,REAL;
  set V = R_Normed_Algebra_of_BoundedFunctions the carrier of ClstoCmp(A);
  defpred P[Nat,set] means
    $2 = sgn(Dp2(rho,D,$1+1) - Dp2(rho,D,$1) )
        * ( Dp1(m,D,$1+1) - Dp1(m,D,$1) );
A0: for i be Nat st i in Seg len D
      ex x be Element of the carrier of V st P[i,x];
  consider s be FinSequence of V such that
A1: dom s = Seg len D
    & for i be Nat st i in Seg len D holds P[i,s.i]
        from FINSEQ_1:sch 5(A0);
  take s;
  thus thesis by A1,FINSEQ_1:def 3;
end;
