
theorem LM95:
  for A be non empty closed_interval Subset of REAL,
      rho be Function of A,REAL,
      B be Real
    st 0 < vol A
  holds
  ( for D be Division of A, K be var_volume of rho,D
      st lower_bound A < D.1
    holds Sum(K) <= B )
  implies
  ( for D be Division of A, K be var_volume of rho,D
    holds Sum(K) <= B )
proof
  let A be non empty closed_interval Subset of REAL,
      rho be Function of A,REAL,
      B be Real;
  assume A1: 0 < vol A;
  assume
A2: for D be Division of A, K be var_volume of rho,D
      st lower_bound A < D.1 holds Sum(K) <= B;
  let D be Division of A, K be var_volume of rho,D;
  1 <= len D by FINSEQ_1:20; then
  1 in dom D by FINSEQ_3:25; then
  D.1 in A by INTEGRA1:6; then
  lower_bound A <= D.1 by SEQ_4:def 2; then
  per cases by XXREAL_0:1;
  suppose
    lower_bound A < D.1;
    hence Sum(K) <= B by A2;
  end;
  suppose A5: lower_bound A = D.1; then
    reconsider E = D /^ 1 as Division of A by A1,LM95A;
A6: lower_bound A < E.1 by A1,A5,LM95B;
    set L = the var_volume of rho,E;
    Sum(L) <= B by A2,A6;
    hence Sum(K) <= B by A1,A5,LM95C;
  end;
end;
