reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

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;
