 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 LMThMBF3:
  for V being finite-rank free Z_Module,
  b1, b2 being OrdBasis of V holds
  AutMt(id(V), b1, b2) is Matrix of rank V,INT.Ring
  proof
    let V be finite-rank free Z_Module,
    b1, b2 be OrdBasis of V;
    set n = rank V;
    A1: len b1 = rank V by ThRank1;
    A2: len b2 = rank V by ThRank1;
    P0: len AutMt(id(V),b1,b2) = len b1 by Def8;
    per cases;
    suppose X1: len b1 = 0;
      then len AutMt(id(V), b1, b2) = 0 by Def8;
      then AutMt(id(V), b1, b2) = {};
      hence thesis by A1,X1,MATRIX_0:13;
    end;
    suppose P1: 0 < len b1; then
      width AutMt(id(V),b1,b2) = len b2 by Th39;
      hence thesis by P0,P1,A1,A2,MATRIX_0:20;
    end;
  end;
