reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem Th40:
  for M being Matrix of D for i,j st [i,j] in Indices M holds (i-1
) * (width M) + j in dom Mx2FinS M & M*(i,j) = (Mx2FinS(M)).((i-1) * (width M)
  + j)
proof
  let M be Matrix of D;
  let i,j such that
A1: [i,j] in Indices M;
  Seg len M <> {} by A1,MATRPROB:12;
  then len M <> 0;
  then
  ex p being FinSequence of D* st Mx2FinS(M) = p.(len M) & len p = len M
  & p.1 = M.1 & for k st k >= 1 & k < len M holds p.(k+1) = (p.k) ^ M.(k+1) by
Def5;
  hence thesis by A1,Th36;
end;
