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 x =
  len M & len y = width M & len y > 0 holds (QuadraticForm(x,M,y))@ =
  QuadraticForm(y,M@,x)
proof
  let x,y be FinSequence of REAL,M be Matrix of REAL;
  assume that
A1: len x = len M and
A2: len y = width M and
A3: len y > 0;
A4: len (M@) = len y by A2,MATRIX_0:def 6;
A5: width (M@) = len x by A1,A2,A3,MATRIX_0:54;
  then
A6: len QuadraticForm(y,M@,x) = len y & width QuadraticForm(y,M@,x) = len x
  by A4,Def4;
A7: len ((QuadraticForm(x,M,y))@) = width (QuadraticForm(x,M,y)) by
MATRIX_0:def 6;
A8: width QuadraticForm(x,M,y)=len y by A1,A2,Def4;
  then
A9: width ((QuadraticForm(x,M,y))@) = len QuadraticForm(x,M,y) by A3,
MATRIX_0:54;
A10: for i,j be Nat st [i,j] in Indices ((QuadraticForm(x,M,y))@) holds (
  QuadraticForm(y,M@,x))*(i,j) = ((QuadraticForm(x,M,y))@)*(i,j)
  proof
    let i,j be Nat;
    assume
A11: [i,j] in Indices ((QuadraticForm(x,M,y))@);
    reconsider i,j as Nat;
A12: 1<=j by A11,MATRIXC1:1;
A13: j<=len QuadraticForm(x,M,y) by A9,A11,MATRIXC1:1;
    then
A14: j<=len M by A1,A2,Def4;
A15: 1<=i by A11,MATRIXC1:1;
A16: i<=width QuadraticForm(x,M,y) by A7,A11,MATRIXC1:1;
    then i<=width M by A1,A2,Def4;
    then
A17: [j,i] in Indices M by A12,A15,A14,MATRIXC1:1;
A18: j<=width (M@) by A1,A2,A5,A13,Def4;
A19: 1<=i by A11,MATRIXC1:1;
    1<=i by A11,MATRIXC1:1;
    then
A20: [j,i] in Indices (QuadraticForm(x,M,y)) by A16,A12,A13,MATRIXC1:1;
    i<=len (M@) by A1,A2,A4,A16,Def4;
    then [i,j] in Indices M@ by A12,A19,A18,MATRIXC1:1;
    then
    (QuadraticForm(y,M@,x))*(i,j) = (y.i)*((M@)*(i,j))*(x.j) by A4,A5,Def4
      .= (y.i)*(M*(j,i))*(x.j) by A17,MATRIX_0:def 6
      .=(x.j)*(M*(j,i))*(y.i)
      .= (QuadraticForm(x,M,y))*(j,i) by A1,A2,A17,Def4
      .= ((QuadraticForm(x,M,y))@)*(i,j) by A20,MATRIX_0:def 6;
    hence thesis;
  end;
A21: len ((QuadraticForm(x,M,y))@) = width (QuadraticForm(x,M,y)) by
MATRIX_0:def 6
    .= len y by A1,A2,Def4;
  width ((QuadraticForm(x,M,y))@) = len QuadraticForm(x,M,y) by A3,A8,
MATRIX_0:54
    .= len x by A1,A2,Def4;
  hence thesis by A21,A6,A10,MATRIX_0:21;
end;
