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

theorem
  for x,y being FinSequence of COMPLEX,M being Matrix of COMPLEX st len
x=len M & len y =width M & width M>0 & len M>0 holds |(x,M*y)| = |((M@")*x,y)|
proof
  let x,y be FinSequence of COMPLEX, M be Matrix of COMPLEX;
  assume that
A1: len x=len M and
A2: len y =width M and
A3: width M>0 and
A4: len M>0;
A5: len ((M@")*x)=len (M@") by Def6
    .=len (M@) by Def1
    .=width M by MATRIX_0:def 6;
  len (M*y) =len x by A1,Def6;
  hence |(x,M*y)| = (|(M*y,x)|)*' by A1,A4,Th52
    .= (|(y,(M@")*x)|)*' by A1,A2,A3,A4,Th57
    .= |((M@")*x,y)| by A2,A3,A5,Th52;
end;
