reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem
  for p being non empty ProbFinS FinSequence of REAL holds Entropy p >= 0
proof
  let p be non empty ProbFinS FinSequence of REAL;
  set p1 = - Infor_FinSeq_of p;
A1: dom p = Seg len p by FINSEQ_1:def 3
    .= Seg len Infor_FinSeq_of p by Th47
    .= dom Infor_FinSeq_of p by FINSEQ_1:def 3;
A2: for k be Nat st k in dom p1 holds p1.k >= 0
  proof
    let k be Nat;
    assume k in dom p1;
    then k in dom p by A1,VALUED_1:8;
    then (Infor_FinSeq_of p).k <= 0 by Th52;
    then -((Infor_FinSeq_of p).k) >= -0;
    hence thesis by RVSUM_1:17;
  end;
  Entropy p = Sum p1 by RVSUM_1:88;
  hence thesis by A2,RVSUM_1:84;
end;
