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 Th44:
  MR is Joint_Probability iff Mx2FinS(MR) is ProbFinS
proof
  hereby
    assume MR is Joint_Probability;
    then reconsider MRR = MR as Joint_Probability Matrix of REAL;
A1: MRR is m-nonnegative with_sum=1 Matrix of REAL;
    then Mx2FinS(MR) is nonnegative by Th43;
    then
A2: for i st i in dom Mx2FinS(MR) holds (Mx2FinS(MR)).i >= 0;
    Sum Mx2FinS(MR) = SumAll MR by Th42
      .= 1 by A1,MATRPROB:def 7;
    hence Mx2FinS(MR) is ProbFinS by A2,MATRPROB:def 5;
  end;
  assume Mx2FinS(MR) is ProbFinS;
  then reconsider pp=Mx2FinS(MR) as ProbFinS FinSequence of REAL;
  reconsider p=pp as non empty ProbFinS FinSequence of REAL;
  SumAll MR = Sum p by Th42
    .= 1 by MATRPROB:def 5;
  then
A3: MR is with_sum=1 by MATRPROB:def 7;
  p is nonnegative;
  then MR is m-nonnegative by Th43;
  hence thesis by A3;
end;
