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

theorem
  for P1, P2 being non empty-yielding Conditional_Probability Matrix of
  REAL st width P1 = len P2 holds P1 * P2 is non empty-yielding
  Conditional_Probability Matrix of REAL & len (P1 * P2) = len P1 & width (P1 *
  P2) = width P2
proof
  let P1, P2 be non empty-yielding Conditional_Probability Matrix of REAL such
  that
A1: width P1 = len P2;
  set n2 = width P2;
  set m = len P2;
  set n1 = len P1;
A2: len(P1 * P2) = n1 by A1,Th42;
A3: width(P1 * P2) = n2 by A1,Th42;
  then reconsider P = P1 * P2 as Matrix of n1,n2,REAL by A2,MATRIX_0:51;
A4: for i st i in dom P holds Line(P,i) is non empty ProbFinS FinSequence of
  REAL
  proof
    let i;
    assume i in dom P;
    then
A5: i in Seg len P by FINSEQ_1:def 3;
    then i in dom P1 by A2,FINSEQ_1:def 3;
    then reconsider
    p = Line(P1,i) as non empty ProbFinS FinSequence of REAL by Th60;
    len p = m by A1,MATRIX_0:def 7;
    then p * P2 is non empty ProbFinS FinSequence of REAL by Th62;
    hence thesis by A1,A5,Th42;
  end;
  n1 > 0 & n2 > 0 by Th54;
  then P is non empty-yielding Matrix of REAL by A2,A3,MATRIX_0:def 10;
  hence thesis by A1,A4,Th42,Th60;
end;
