
theorem ThMBFY:
  for V being finite-rank free Z_Module,
      b1, b2 being OrdBasis of V, f being bilinear-FrForm of V, V
    holds Det BilinearM(f, b2, b2) = Det BilinearM(f, b1, b1)
  proof
    let V be finite-rank free Z_Module,
    b1, b2 be OrdBasis of V, f be bilinear-FrForm of V, V;
    set n = len b1;
    A1: len b1 = rank V by ZMATRLIN:49;
    A2: len b2 = rank V by ZMATRLIN:49;
    reconsider B1 = BilinearM(f, b1, b1) as Matrix of n,F_Real;
    reconsider B2 = BilinearM(f, b2, b2) as Matrix of n,F_Real by A1,A2;
    reconsider I1 = AutMt(id(V), b2, b1) as Matrix of n,INT.Ring
    by A1,ZMATRLIN:50;
    set I = inttorealM(I1);
    per cases;
    suppose rank V = 0;
      hence Det BilinearM(f, b2, b2) = Det BilinearM(f, b1, b1)
      by A1,A2,MATRIX_0:45;
    end;
    suppose ZZ: rank V > 0;
      then B2: BilinearM(f, b2, b2) = I * (BilinearM(f, b1, b1)) * (I@)
      by ThMBF3;
      set M1 = I@;
      set M2 = I;
      width M2 = n & len M2 = n &
      width B1 = n & len B1 = n by MATRIX_0:24;
      then width M2 = len B1 & len M2 > 0 & len B1> 0 by ZZ,ZMATRLIN:49;
      then reconsider M2B1 = M2 * B1 as Matrix of n, F_Real by MATRIX_4:64;
      X2: (Det M2 = 1 & Det M1 = 1) or (Det M2 = -1 & Det M1 = -1)
      by A1,ThSign2;
      thus Det BilinearM(f, b2, b2) = Det B2 by A1,ZMATRLIN:49
      .= (Det(M2B1)) * (Det M1) by A1,B2,ZZ,MATRIX11:62
      .= ((Det M2) * (Det B1)) * (Det M1) by ZZ,A1,MATRIX11:62
      .= Det BilinearM(f, b1, b1) by X2;
    end;
  end;
