theorem
  f1 is additive homogeneous & f2 is additive homogeneous &
   AutMt(f1,b1,b2) = AutMt(f2,b1,b2) & len b1 > 0 implies f1 = f2
proof
  assume that
A1: f1 is additive homogeneous & f2 is additive homogeneous and
A2: AutMt(f1,b1,b2) = AutMt(f2,b1,b2) and
A3: len b1 > 0;
  rng b1 is Basis of V1 by Def2;
  then
A4: rng b1 c= the carrier of V1;
  then
A5: rng b1 c= dom f2 by FUNCT_2:def 1;
  rng b1 c= dom f1 by A4,FUNCT_2:def 1;
  then
A6: dom (f1*b1) = dom b1 by RELAT_1:27
    .= dom (f2*b1) by A5,RELAT_1:27;
  now
    let x be object;
    assume
A7: x in dom (f1*b1);
    then reconsider k=x as Nat by FINSEQ_3:23;
A8: dom (f1*b1) c= dom b1 by RELAT_1:25;
    then
A9: f1.(b1/.k) |-- b2 = (AutMt(f2,b1,b2))/.k by A2,A7,Def8
      .= f2.(b1/.k) |-- b2 by A7,A8,Def8;
    thus (f1*b1).x = f1.(b1.x) by A7,FUNCT_1:12
      .= f1.(b1/.x) by A7,A8,PARTFUN1:def 6
      .= f2.(b1/.x) by A9,Th34
      .= f2.(b1.x) by A7,A8,PARTFUN1:def 6
      .= (f2*b1).x by A6,A7,FUNCT_1:12;
  end;
  hence thesis by A1,A3,A6,Th22,FUNCT_1:2;
end;
