reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th57:
  for x,y being FinSequence of COMPLEX, M being Matrix of COMPLEX
st len x=width M & len y =len M & width M>0 & len M>0 holds |(M*x,y)| = |(x,(M
  @")*y)|
proof
  let x,y be FinSequence of COMPLEX, M be Matrix of COMPLEX;
  assume that
A1: len x=width M and
A2: len y =len M and
A3: width M>0 and
A4: len M >0;
  |(x,(M@")*y)| = SumAll(QuadraticForm(x,M@,y)) by A1,A2,A3,A4,Th55;
  hence thesis by A1,A2,A3,A4,Th56;
end;
