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 Th31:
  len b1>0 & f is additive homogeneous implies LineVec2Mx(v1|-- b1)*AutMt(f,b1,
  b2) = LineVec2Mx (f.v1 |-- b2)
proof
  assume that
A1: len b1>0 and
A2: f is additive homogeneous;
  set A=AutMt(f,b1,b2);
  set fb=f.v1 |-- b2;
  set vb=v1|-- b1;
  set L=LineVec2Mx vb;
  set LA=L*A;
  set Lf=LineVec2Mx fb;
A3: len A=len b1 by MATRLIN:def 8;
  len fb=len b2 by MATRLIN:def 7;
  then
A4: width Lf=len b2 by MATRIX_0:23;
A5: len vb=len b1 by MATRLIN:def 7;
  then
A6: width L=len b1 by MATRIX_0:23;
  len L=1 by MATRIX_0:23;
  then
A7: len LA=1 by A6,A3,MATRIX_3:def 4;
A8: width A = len b2 by A1,MATRLIN:39;
  then
A9: width LA=len b2 by A6,A3,MATRIX_3:def 4;
A10: now
A11: dom b2=Seg len b2 by FINSEQ_1:def 3;
A12: dom LA=Seg 1 by A7,FINSEQ_1:def 3;
A13: len (f*b1)=len b1 by FINSEQ_2:33;
    let i,j such that
A14: [i,j] in Indices LA;
A15: j in Seg len b2 by A9,A14,ZFMISC_1:87;
    i in dom LA by A14,ZFMISC_1:87;
    then
A16: i=1 by A12,FINSEQ_1:2,TARSKI:def 1;
A17: len Col(A,j)=len A by CARD_1:def 7;
A18: now
A19:  dom (f*b1)=dom b1 by A13,FINSEQ_3:29;
A20:  dom A=dom Col(A,j) by A17,FINSEQ_3:29;
      let k such that
A21:  k in dom Col(A,j);
A22:  dom A=Seg len A & A.k=A/.k by A21,A20,FINSEQ_1:def 3,PARTFUN1:def 6;
A23:  dom A=dom b1 by A3,FINSEQ_3:29;
      then
A24:  f.(b1/.k) = f.(b1.k) by A21,A20,PARTFUN1:def 6
        .= (f*b1).k by A21,A20,A23,FUNCT_1:13
        .= (f*b1)/.k by A21,A20,A23,A19,PARTFUN1:def 6;
      thus Col(A,j).k = A*(k,j) by A21,A20,MATRIX_0:def 8
        .= Line(A,k).j by A8,A15,MATRIX_0:def 7
        .= (A/.k).j by A3,A21,A20,A22,MATRIX_0:52
        .= ((f*b1)/.k|--b2).j by A21,A20,A23,A24,MATRLIN:def 8;
    end;
    thus Lf*(i,j) = Line(Lf,i).j by A4,A15,MATRIX_0:def 7
      .= (f.v1 |-- b2).j by A16,MATRIX15:25
      .= (f.(Sum(lmlt(v1|--b1,b1))) |--b2).j by MATRLIN:35
      .= (Sum lmlt(v1|--b1,f*b1) |--b2).j by A2,A5,MATRLIN:18
      .= (v1|-- b1)"*"Col(A,j) by A1,A5,A3,A11,A15,A13,A17,A18,Th30
      .= Line(L,1)"*"Col(A,j) by MATRIX15:25
      .= LA*(i,j) by A6,A3,A14,A16,MATRIX_3:def 4;
  end;
  len Lf=1 by MATRIX_0:23;
  hence thesis by A7,A9,A4,A10,MATRIX_0:21;
end;
