reserve x, y for object, X for set,
  i, j, k, l, n, m for Nat,
  D for non empty set,
  K for commutative Ring,
  a,b for Element of K,
  perm, p, q for Element of Permutations(n),
  Perm,P for Permutation of Seg n,
  F for Function of Seg n,Seg n,
  perm2, p2, q2, pq2 for Element of Permutations(n+2),
  Perm2 for Permutation of Seg (n+2);
reserve s for Element of 2Set Seg (n+2);
reserve pD for FinSequence of D,
  M for Matrix of n,m,D,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem Th38:
  for M be Matrix of n,m,D,F for k st k in Seg n holds Line(M*F,k) = M.(F.k)
proof
  let M be Matrix of n,m,D,F;
  let k such that
A1: k in Seg n;
  len (M*F)=n by MATRIX_0:def 2;
  then k in dom (M*F) by A1,FINSEQ_1:def 3;
  then (M*F).k=M.(F.k) by FUNCT_1:12;
  hence thesis by A1,MATRIX_0:52;
end;
