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 Th52:
  for p being non empty ProbFinS FinSequence of REAL for k st k in
  dom p holds (Infor_FinSeq_of p).k <= 0
proof
  let p be non empty ProbFinS FinSequence of REAL, k such that
A1: k in dom p;
  per cases by A1,MATRPROB:53,def 5;
  suppose
    p.k = 0;
    hence thesis by A1,Th48;
  end;
  suppose
A2: p.k > 0 & p.k <= 1;
    then
A3: (Infor_FinSeq_of p).k = p.k * log(2,p.k) by A1,Th48;
    thus (Infor_FinSeq_of p).k <= 0
    proof
A4:   log(2,1) = 0 by POWER:51;
      per cases by A2,XXREAL_0:1;
      suppose
        p.k < 1;
        then log(2,p.k) < 0 by A2,A4,POWER:57;
        hence thesis by A2,A3;
      end;
      suppose
        p.k = 1;
        hence thesis by A3,POWER:51;
      end;
    end;
  end;
end;
