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 Th63:
  for M being non empty-yielding Conditional_Probability Matrix of
  REAL holds Entropy_of_Cond_Prob M = -LineSum Infor_FinSeq_of M
proof
  let M be non empty-yielding Conditional_Probability Matrix of REAL;
  set p=Entropy_of_Cond_Prob M;
  set q=-LineSum Infor_FinSeq_of M;
A1: dom q = dom LineSum Infor_FinSeq_of M by VALUED_1:8;
  then len q = len LineSum Infor_FinSeq_of M by FINSEQ_3:29;
  then len q = len Infor_FinSeq_of M by MATRPROB:def 1;
  then
A2: len q = len M by Def8;
  len p = len M by Th62;
  then
A3: dom p = dom q by A2,FINSEQ_3:29;
  now
    let k be Nat such that
A4: k in dom p;
    thus p.k = -Sum ((Infor_FinSeq_of M).k) by A4,Th62
      .= -(Sum Infor_FinSeq_of M).k by A1,A3,A4,MATRPROB:def 1
      .= q.k by RVSUM_1:17;
  end;
  hence thesis by A3,FINSEQ_1:13;
end;
