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

theorem Th51:
  for x,y being FinSequence of COMPLEX,M being Matrix of COMPLEX
st len x=len M & len y =width M holds (QuadraticForm(x,M,y))*'=QuadraticForm(x
  *',M*',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;
A3: len (y*') = len y by COMPLSP2:def 1;
  then
A4: len (y*') = width (M*') by A2,Def1;
  len (M*') = len M by Def1;
  then
A5: len (x*') = len (M*') by A1,COMPLSP2:def 1;
  then
A6: len (QuadraticForm(x*',M*',y*')) = len (x*') by A4,Def12
    .= len M by A1,COMPLSP2:def 1;
A7: width ((QuadraticForm(x,M,y))*')=width QuadraticForm(x,M,y) by Def1;
A8: len ((QuadraticForm(x,M,y))*') = len QuadraticForm(x,M,y) by Def1;
A9: for i,j st [i,j] in Indices ((QuadraticForm(x,M,y))*') holds ((
  QuadraticForm(x,M,y))*')*(i,j) = (QuadraticForm(x*',M*',y*'))*(i,j)
  proof
    let i,j;
    reconsider i9=i, j9=j as Element of NAT by ORDINAL1:def 12;
    assume
A10: [i,j] in Indices ((QuadraticForm(x,M,y))*');
    then
A11: 1<=i by Th1;
A12: i<=len (QuadraticForm(x,M,y)) by A8,A10,Th1;
    then
A13: i<=len x by A1,A2,Def12;
    i<=len M by A1,A2,A12,Def12;
    then
A14: i<=len (M*') by Def1;
A15: j<=width (QuadraticForm(x,M,y)) by A7,A10,Th1;
    then
A16: j<=len y by A1,A2,Def12;
    j<=width M by A1,A2,A15,Def12;
    then
A17: j<=width (M*') by Def1;
    1<=j by A10,Th1;
    then
A18: [i,j] in Indices (M*') by A11,A14,A17,Th1;
A19: j<=width M by A1,A2,A15,Def12;
A20: 1<=i by A10,Th1;
A21: 1<=j by A10,Th1;
    i<=len M by A1,A2,A12,Def12;
    then
A22: [i,j] in Indices M by A21,A20,A19,Th1;
    [i,j] in Indices QuadraticForm(x,M,y) by A11,A12,A21,A15,Th1;
    then
    ((QuadraticForm(x,M,y))*')*(i,j) =((QuadraticForm(x,M,y))*(i,j))*' by Def1
      .=((x.i)*(M*(i,j))*((y.j)*'))*' by A1,A2,A22,Def12
      .=((x.i)*(M*(i9,j9))*((y*'.j)))*' by A21,A16,COMPLSP2:def 1
      .= (((x.i)*(M*(i,j)))*')*((y*'.j)*') by COMPLEX1:35
      .= (((x.i)*(M*(i9,j9)))*')*((y*')*'.j) by A3,A21,A16,COMPLSP2:def 1
      .= ((x.i)*')*((M*(i,j)*'))*(((y*')*'.j)) by COMPLEX1:35
      .= ((x.i)*')*((M*')*(i,j))*(((y*')*'.j)) by A22,Def1
      .= ((x.i)*')*((M*')*(i9,j9))*(((y*').j)*') by A3,A21,A16,COMPLSP2:def 1
      .= ((x*').i)*((M*')*(i9,j9))*(((y*').j)*') by A11,A13,COMPLSP2:def 1
      .= (QuadraticForm(x*',M*',y*'))*(i,j) by A5,A4,A18,Def12;
    hence thesis;
  end;
A23: width ((QuadraticForm(x,M,y))*')=width (QuadraticForm(x,M,y)) by Def1
    .= len y by A1,A2,Def12;
A24: width QuadraticForm(x*',M*',y*')=len (y*') by A5,A4,Def12
    .= len y by COMPLSP2:def 1;
  len ((QuadraticForm(x,M,y))*') = len QuadraticForm(x,M,y) by Def1
    .= len M by A1,A2,Def12;
  hence thesis by A6,A23,A24,A9,MATRIX_0:21;
end;
