
theorem ThMBF3:
  for V being finite-rank free Z_Module,
  b1, b2 being OrdBasis of V, f being bilinear-FrForm of V,V
  st 0 < rank V holds
  BilinearM(f, b2, b2)
  = inttorealM( AutMt(id(V), b2, b1))  * BilinearM(f, b1, b1)
  * ((inttorealM( AutMt(id(V), b2, b1)))@)
  proof
    let V be finite-rank free Z_Module,
    b1, b2 be OrdBasis of V, f be bilinear-FrForm of V,V;
    set I = inttorealM( AutMt(id(V), b2, b1));
    assume AS: 0 < rank V;
    set n = len b1;
    A1: len b1 = rank V by ZMATRLIN:49;
    reconsider IM1 = AutMt(id(V), b2, b1) as Matrix of
    n,INT.Ring by ZMATRLIN:50,A1;
    reconsider IM2 = AutMt(id(V), b2, b1) as Matrix of n,INT.Ring
    by ZMATRLIN:50,A1;
    reconsider M1 = IM1@ as Matrix of n,INT.Ring;
    reconsider M2 = IM2 as Matrix of n,INT.Ring;
    Y1: width IM1=n by MATRIX_0:24;
    Yb: width BilinearM(f, b1, b1)=len b1 by MATRIX_0:24;
    width (AutMt(id(V), b2, b1))=len BilinearM(f, b1, b1) &
    width BilinearM(f, b1, b1)=len (AutMt(id(V), b2, b1)@)
    by MATRIX_0:def 2,Y1,Yb;
    then
    X1: width I=len BilinearM(f, b1, b1) &
    width BilinearM(f, b1, b1)=len (I@) by ZMATRLIN:6;
    thus BilinearM(f, b2, b2)
    = I* BilinearM(f, b1, b2) by ThMBF2,AS
    .= I* ( BilinearM(f, b1, b1) * (I@)) by ThMBF1,AS
    .= I * BilinearM(f, b1, b1) *(I@) by MATRIX_3:33,X1;
  end;
