reserve k,t,i,j,m,n for Nat,
  x,y,y1,y2 for object,
  D for non empty set;
reserve K for Field,
  V for VectSp of K,
  a for Element of K,
  W for Element of V;
reserve KL1,KL2,KL3 for Linear_Combination of V,
  X for Subset of V;
reserve s for FinSequence,
  V1,V2,V3 for finite-dimensional VectSp of K,
  f,f1,f2 for Function of V1,V2,
  g for Function of V2,V3,
  b1 for OrdBasis of V1,
  b2 for OrdBasis of V2,
  b3 for OrdBasis of V3,
  v1,v2 for Vector of V2,
  v,w for Element of V1;
reserve p2,F for FinSequence of V1,
  p1,d for FinSequence of K,
  KL for Linear_Combination of V1;

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;
