 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
  for V being finite-rank free Z_Module, b1, b2 being OrdBasis of V,
  M being Matrix of rank(V),F_Real st M = AutMt(id(V), b1, b2)
  holds Det M in INT
  proof
    let V be finite-rank free Z_Module,
    b1, b2 be OrdBasis of V,
    M be Matrix of rank(V),F_Real;
    assume A2: M = AutMt(id(V), b1, b2);
    per cases;
    suppose not 0 < rank(V);
      then rank(V) = 0;
      then Det M = 1.F_Real by MATRIXR2:41;
      hence Det M in INT;
    end;
    suppose A3: 0 < rank(V);
      len M = rank(V) & width M = rank(V) by MATRIX_0:24;
      then M is Matrix of rank(V),INT by A2,A3,MATRIX_0:20;
      hence thesis by LmSign1A;
    end;
  end;
