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 Th48:
  for p being nonnegative FinSequence of REAL for k st k in dom p
  holds (p.k = 0 implies (Infor_FinSeq_of p).k = 0) & (p.k > 0 implies (
  Infor_FinSeq_of p).k = p.k * log(2,p.k))
proof
  let p be nonnegative FinSequence of REAL;
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;
  let k such that
A2: k in dom p;
  hereby
    assume p.k=0;
    hence (Infor_FinSeq_of p).k = 0*log(2,p.k) by A2,A1,Th47
      .=0;
  end;
  thus thesis by A2,A1,Th47;
end;
