 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   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-rank free Z_Module,
   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 INT.Ring,
   KL for Linear_Combination of V1;

theorem LmSign3:
  for V being finite-rank free Z_Module, b1, b2 being OrdBasis of V
  st rank(V) > 0
  holds (AutMt(id(V), b1, b2)) * (AutMt(id(V), b2, b1)) = 1.(INT.Ring,rank(V))
  proof
    let V be finite-rank free Z_Module, b1, b2 be OrdBasis of V;
    assume AS: rank(V) > 0;
    then A1: len b1 > 0 & len b2 > 0 by ThRank1;
    thus (AutMt(id(V), b1, b2))
    * (AutMt(id(V), b2, b1)) =
    AutMt(id(V)*id(V), b1, b1) by A1,ThComp1
    .= AutMt(id(V), b1, b1) by FUNCT_2:17
    .= 1.(INT.Ring,rank(V)) by AS,LmSign31;
  end;
