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,q being ProbFinS FinSequence of REAL holds Entropy_of_Joint_Prob
  ((ColVec2Mx p) * (LineVec2Mx q)) = Entropy p + Entropy q
proof
  let p,q be ProbFinS FinSequence of REAL;
  set M = (ColVec2Mx p) * (LineVec2Mx q);
  reconsider M as Joint_Probability Matrix of REAL by Th23;
  set pp = Mx2FinS((ColVec2Mx p) * (LineVec2Mx q));
  reconsider pp as ProbFinS FinSequence of REAL by Th45;
  Entropy p + Entropy q = -(Sum Infor_FinSeq_of p + Sum Infor_FinSeq_of q)
    .= -(SumAll (Infor_FinSeq_of M)) by Th60
    .= -(Sum (Mx2FinS(Infor_FinSeq_of M))) by Th42
    .= -(Sum (Infor_FinSeq_of pp)) by Th59;
  hence thesis;
end;
