reserve D for non empty set,
  i,j,k for Nat,
  n,m for Nat,
  r for Real,
  e for real-valued FinSequence;

theorem Th60:
  for M being non empty-yielding Matrix of REAL holds M is non
  empty-yielding Conditional_Probability Matrix of REAL iff for i st i in dom M
  holds Line(M,i) is non empty ProbFinS FinSequence of REAL
proof
  let M be non empty-yielding Matrix of REAL;
  hereby
    assume
A1: M is non empty-yielding Conditional_Probability Matrix of REAL;
    hereby
      set m = width M;
      let i such that
A2:   i in dom M;
      for i,j st [i,j] in Indices M holds M*(i,j) >=0 by A1,Def6;
      then
A3:   len Line(M,i) = m & for j st j in dom Line(M,i) holds Line(M,i).j >=
      0 by A2,Lm2,MATRIX_0:def 7;
      Sum Line(M,i) = Sum (M.i) by A2,MATRIX_0:60
        .= 1 by A1,A2,Def9;
      hence Line(M,i) is non empty ProbFinS FinSequence of REAL by A3,Def5,Th54
;
    end;
  end;
  assume
A4: for i st i in dom M holds Line(M,i) is non empty ProbFinS
  FinSequence of REAL;
  now
    let k such that
A5: k in dom M;
    Line(M,k) is ProbFinS by A4,A5;
    then Sum Line(M,k) = 1;
    hence Sum(M.k) = 1 by A5,MATRIX_0:60;
  end;
  then
A6: M is with_line_sum=1;
  for i,j st i in dom M & j in dom Line(M,i) holds Line(M,i).j >=0
  proof
    let i,j such that
A7: i in dom M and
A8: j in dom Line(M,i);
    Line(M,i) is ProbFinS by A4,A7;
    hence thesis by A8;
  end;
  then for i,j st [i,j] in Indices M holds M*(i,j) >=0 by Lm2;
  then M is m-nonnegative;
  hence thesis by A6;
end;
