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

theorem Th56:
  for M being non empty-yielding Joint_Probability Matrix of REAL
  holds M@ is non empty-yielding Joint_Probability Matrix of REAL
proof
  let M be non empty-yielding Joint_Probability Matrix of REAL;
  set n = len M;
  set m = width M;
A1: n > 0 by Th54;
A2: m > 0 by Th54;
  then
A3: len (M@) = m & width (M@) = n by MATRIX_0:54;
  then reconsider M1=M@ as Matrix of m,n,REAL by MATRIX_0:51;
  for i,j st [i,j] in Indices M1 holds M1*(i,j) >=0
  proof
    let i,j;
    assume [i,j] in Indices M1;
    then
A4: [j,i] in Indices M by MATRIX_0:def 6;
    then M1*(i,j) = M*(j,i) by MATRIX_0:def 6;
    hence thesis by A4,Def6;
  end;
  then
A5: M1 is m-nonnegative;
  SumAll M1 = SumAll M by Th28
    .= 1 by Def7;
  then M1 is with_sum=1;
  hence thesis by A1,A2,A3,A5,MATRIX_0:def 10;
end;
