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,y being FinSequence of REAL,M being Matrix of REAL st len y=len
  M & len x =width M & len x>0 & len y>0 holds |((M*x),y)| = |(x,(M@*y))|
proof
  let x,y be FinSequence of REAL, M be Matrix of REAL such that
A1: len y=len M and
A2: len x =width M and
A3: len x>0 and
A4: len y>0;
A5: len (M@) = width M & width (M@) = len M by A2,A3,MATRIX_0:54;
  thus |((M*x),y)| = |((x*M@),y)| by A1,A2,A3,A4,MATRIXR1:53
    .= |(x,(M@*y))| by A1,A2,A4,A5,Th47;
end;
