theorem Th34:
  f is additive homogeneous implies Mx2Tran(AutMt(f,b1,b2),b1,b2) = f
proof
  assume
A1: f is additive homogeneous;
  set A=AutMt(f,b1,b2);
  set M=Mx2Tran(A,b1,b2);
  per cases;
  suppose
A2: len b1=0;
    now
A3:   b1 is one-to-one by MATRLIN:def 2;
      reconsider R=rng b1 as Basis of V1 by MATRLIN:def 2;
      let x be object such that
A4:   x in the carrier of V1;
A5:    Seg len b1 = {} by A2;
      dim V1 = card R by VECTSP_9:def 1
        .= card dom b1 by A3,CARD_1:70
        .= 0 by A5,CARD_1:27,FINSEQ_1:def 3;
      then (Omega).V1 = (0).V1 by VECTSP_9:29;
      then the carrier of V1 = {0.V1} by VECTSP_4:def 3;
      then x=0.V1 by A4,TARSKI:def 1;
      hence f.x = f.(0.K*0.V1) by VECTSP_1:15
        .= 0.K*(f.(0.V1)) by A1,MOD_2:def 2
        .= 0.V2 by VECTSP_1:15
        .= M.x by A2,A4,Th33;
    end;
    hence thesis by FUNCT_2:12;
  end;
  suppose
A6: len b1>0;
    reconsider fb=f*b1,Mf=M*b1 as FinSequence;
A7: rng b1 is Subset of V1 by FINSEQ_1:def 4;
    dom f=the carrier of V1 by FUNCT_2:def 1;
    then
A8: len fb=len b1 by A7,FINSEQ_2:29;
    dom M=the carrier of V1 by FUNCT_2:def 1;
    then
A9: len Mf=len b1 by A7,FINSEQ_2:29;
    now
A10:   dom fb=dom Mf by A8,A9,FINSEQ_3:29;
      let i;
      assume 1<=i & i<=len fb;
      then
A11:  i in dom fb by FINSEQ_3:25;
      dom fb=dom b1 by A8,FINSEQ_3:29;
      then
A12:  b1.i = b1/.i by A11,PARTFUN1:def 6;
      LineVec2Mx(M.(b1/.i) |--b2) = LineVec2Mx(b1/.i|--b1) * A by A6,Th32
        .= LineVec2Mx (f.(b1/.i) |-- b2) by A1,A6,Th31;
      then M.(b1/.i) |--b2 = Line(LineVec2Mx(f.(b1/.i) |--b2),1) by MATRIX15:25
        .= f.(b1/.i) |--b2 by MATRIX15:25;
      then M.(b1/.i)=f.(b1/.i) by MATRLIN:34;
      hence fb.i = M.(b1/.i) by A11,A12,FUNCT_1:12
        .= Mf.i by A11,A10,A12,FUNCT_1:12;
    end;
    hence thesis by A1,A6,A8,A9,FINSEQ_1:14,MATRLIN:22;
  end;
end;
