theorem Th47:
  for M be OrdBasis of (len b2)-VectSp_over K st 
  M = MX2FinS 1.(K,len b2) 
  for A be Matrix of len b1,len M,K st A = AutMt(f,b1,b2) &
  f is additive homogeneous
  holds Mx2Tran(A,b1,M).v1 = f.v1 |-- b2
proof
  let M be OrdBasis of (len b2)-VectSp_over K such that
A1: M = MX2FinS 1.(K,len b2);
  let A be Matrix of len b1,len M,K such that
A2: A = AutMt(f,b1,b2) and
A3: f is additive homogeneous;
  reconsider f9=f as linear-transformation of V1,V2 by A3;
  set MM=Mx2Tran(A,b1,M);
  per cases;
  suppose
A4: len b1=0;
    then dim V1=0 by Th21;
    then (Omega).V1=(0).V1 by VECTSP_9:29;
    then the carrier of V1={0.V1} by VECTSP_4:def 3;
    then v1=0.V1 by TARSKI:def 1;
    then v1 in ker f9 by RANKNULL:11;
    hence f.v1|--b2 = 0.V2|--b2 by RANKNULL:10
      .= len b2|-> 0.K by Th20
      .= 0.((len b2)-VectSp_over K) by MATRIX13:102
      .= MM.v1 by A4,Th33;
  end;
  suppose
A5: len b1>0;
    then LineVec2Mx(MM.v1|--M) = LineVec2Mx(v1|--b1)*A by Th32
      .= LineVec2Mx(f.v1|--b2) by A2,A3,A5,Th31;
    hence f.v1|--b2 = Line(LineVec2Mx(MM.v1|--M),1) by MATRIX15:25
      .= MM.v1|--M by MATRIX15:25
      .= MM.v1 by A1,Th46;
  end;
end;
