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

theorem Th50:
  for x being FinSequence of REAL,M being Matrix of REAL st len x
= len M & x = len x |-> 1 holds for k st k in Seg len(x*M) holds (x*M).k = Sum
  Col(M,k)
proof
  let x be FinSequence of REAL, M be Matrix of REAL such that
A1: len x = len M and
A2: x = len x |-> 1;
  hereby
    let k such that
A3: k in Seg len(x*M);
A4: len Col(M,k) = len x by A1,MATRIX_0:def 8;
    thus (x*M).k = x "*" Col(M,k) by A1,A3,Th40
      .= Sum Col(M,k) by A2,A4,Th32;
  end;
end;
