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 M being non empty-yielding Conditional_Probability Matrix of REAL
  holds for i,j st [i,j] in Indices M holds M*(i,j) <= 1
proof
  let M be non empty-yielding Conditional_Probability Matrix of REAL;
A1: for i,j st [i,j] in Indices M holds M*(i,j) >= 0 by Def6;
A2: for i st i in dom M holds for j st j in dom(M.i) holds (M.i).j <= 1
  proof
    let i such that
A3: i in dom M;
A4: for j be Nat st j in dom(M.i) holds (M.i).j >= 0 by A1,A3,Lm1;
    let j;
    assume j in dom(M.i);
    then (M.i).j <= Sum(M.i) by A4,Th5;
    hence thesis by A3,Def9;
  end;
  let i,j such that
A5: [i,j] in Indices M;
A6: ex p1 being FinSequence of REAL st p1 = M.i & M*(i,j) = p1.j by A5,
MATRIX_0:def 5;
  i in Seg len M by A5,Th12;
  then
A7: i in dom M by FINSEQ_1:def 3;
  j in Seg width M by A5,Th12;
  then j in Seg len(M.i) by A7,MATRIX_0:36;
  then j in dom(M.i) by FINSEQ_1:def 3;
  hence thesis by A2,A7,A6;
end;
