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 x being FinSequence of REAL,M being Matrix of REAL st len x =
  width M & width M > 0 & x = (len x |-> 1) holds for k st k in Seg len(M*x)
  holds (M*x).k = Sum Line(M,k)
proof
  let x be FinSequence of REAL, M be Matrix of REAL such that
A1: len x = width M and
A2: width M > 0 and
A3: x = (len x |-> 1);
  hereby
    let k such that
A4: k in Seg len(M*x);
A5: len Line(M,k) = len x by A1,MATRIX_0:def 7;
    thus (M*x).k =Line(M,k)"*" x by A1,A2,A4,Th41
      .= Sum Line(M,k) by A3,A5,Th32;
  end;
end;
