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

theorem Th56:
  for x,y being FinSequence of COMPLEX,M being Matrix of COMPLEX
  st len y=len M & len x =width M & len x>0 & len y>0 holds |(M*x,y)| = SumAll
  QuadraticForm(x,M@,y)
proof
  let x,y be FinSequence of COMPLEX, M be Matrix of COMPLEX;
  assume that
A1: len y=len M and
A2: len x =width M and
A3: len x>0 and
A4: len y>0;
A5: (M@)@"=M*' by A1,A2,A3,A4,MATRIX_0:57;
A6: width (M@)=len M by A2,A3,MATRIX_0:54;
A7: len (x*')= width M by A2,COMPLSP2:def 1;
  len (y*')= len M by A1,COMPLSP2:def 1;
  then
A8: len QuadraticForm(y*',M,x*') = len M by A7,Def12;
A9: len (x*')=len x by COMPLSP2:def 1;
A10: 0+1<=len x by A3,NAT_1:13;
A11: len (y*')=len y by COMPLSP2:def 1;
A12: len (M@)=width M by MATRIX_0:def 6;
A13: len (M*x)=len M by Def6;
  hence |(M*x,y)| = (|(y,M*x)|)*' by A1,A4,Th52
    .= (|(y*',(M*x)*')|) by A1,A4,A13,Th53
    .= |(y*',((M*')*(x*')) )| by A2,A10,Th41
    .= SumAll QuadraticForm(y*',(M@)@,x*') by A1,A2,A3,A4,A9,A11,A12,A6,A5,Th55
    .= SumAll QuadraticForm(y*',M,x*') by A1,A2,A3,A4,MATRIX_0:57
    .= SumAll((QuadraticForm(y*',M,x*'))@) by A1,A4,A8,Th49
    .= SumAll((QuadraticForm(x*',(M@"),y*'))*') by A1,A2,A3,A9,A11,Th50
    .= SumAll(((QuadraticForm(x,(M@),y))*')*') by A1,A2,A12,A6,Th51
    .= SumAll QuadraticForm(x,M@,y);
end;
