
theorem LM1:
for A be non empty closed_interval Subset of REAL,
    rho be Function of A,REAL,
    t be Division of A,
    F be var_volume of rho,t
 holds for k be Nat st k in dom F holds 0 <= F.k
proof
  let A be non empty closed_interval Subset of REAL,
      rho be Function of A,REAL,
      t be Division of A,
      F be var_volume of rho,t;
  let k be Nat;
  assume k in dom F; then
  k in Seg len F by FINSEQ_1:def 3; then
  k in Seg len t by defvm; then
  k in dom t by FINSEQ_1:def 3; then
  F.k = |. vol (divset(t,k),rho) .| by defvm;
  hence thesis by COMPLEX1:46;
end;
