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 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;
