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

theorem Th62:
  for p being non empty ProbFinS FinSequence of REAL for P being
  non empty-yielding Conditional_Probability Matrix of REAL st len p = len P
  holds p * P is non empty ProbFinS FinSequence of REAL & len (p * P) = width P
proof
  let p be non empty ProbFinS FinSequence of REAL;
  set n = len p;
  let P be non empty-yielding Conditional_Probability Matrix of REAL such that
A1: len p = len P;
A2: len(p * P) = width P by A1,MATRIXR1:62;
A3: for i st i in dom (p*P) holds (p*P).i >= 0
  proof
    let i such that
A4: i in dom(p*P);
    i in Seg len (p*P) by A4,FINSEQ_1:def 3;
    then
A5: (p*P).i = p "*" Col(P,i) by A1,Th40
      .= Sum(mlt(p,Col(P,i)));
A6: for i,j st [i,j] in Indices P holds P*(i,j) >=0 by Def6;
    i in Seg width P by A2,A4,FINSEQ_1:def 3;
    then
A7: for j st j in dom Col(P,i) holds (Col(P,i)).j >=0 by A6,Lm3;
    for i st i in dom p holds p.i >= 0 by Def5;
    then
    for k be Nat st k in dom mlt(p,Col(P,i)) holds (mlt(p,Col(P,i))).k >=
    0 by A7,Th33;
    hence thesis by A5,RVSUM_1:84;
  end;
  set m = width P;
  set e = m |-> jj;
A8: len e = m by CARD_1:def 7;
A9: m > 0 by Th54;
  then
A10: len(P@) = width P by MATRIX_0:54;
  width(P@) = len P by A9,MATRIX_0:54;
  then
A11: len(e*P@) = n by A1,A8,A10,MATRIXR1:62;
  for k be Nat st k in dom(e*P@) holds (e*P@).k = 1
  proof
    let k be Nat;
    assume k in dom(e*P@);
    then
A12: k in Seg len(e*P@) by FINSEQ_1:def 3;
    then
A13: k in dom P by A1,A11,FINSEQ_1:def 3;
    thus (e*P@).k = Sum Col(P@,k) by A8,A10,A12,Th50
      .= Sum Line(P,k) by A13,MATRIX_0:58
      .= Sum (P.k) by A13,MATRIX_0:60
      .= 1 by A13,Def9;
  end;
  then
A14: e*P@ = n |-> jj by A11,Th1;
  Sum(p*P) = |((p*P),e)| by A2,Th32
    .= |(p,e*P@)| by A1,A9,A8,Th49
    .= Sum p by A14,Th32
    .= 1 by Def5;
  hence thesis by A2,A3,Def5,Th54;
end;
