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;
