 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 Th39:
  len b1 > 0 implies width AutMt(f,b1,b2) = len b2
  proof
    assume len b1 > 0;
    then len AutMt(f,b1,b2) > 0 by Def8;
    then consider s be FinSequence such that
    A1: s in rng AutMt(f,b1,b2) and
    A2: len s = width AutMt(f,b1,b2) by MATRIX_0:def 3;
    consider i be Nat such that
    A3: i in dom AutMt(f,b1,b2) and
    A4: AutMt(f,b1,b2).i = s by A1,FINSEQ_2:10;
    len AutMt(f,b1,b2) = len b1 by Def8;
    then
    A5: i in dom b1 by A3,FINSEQ_3:29;
    s = (AutMt(f,b1,b2))/.i by A3,A4,PARTFUN1:def 6
    .= f.(b1/.i) |-- b2 by A5,Def8;
    hence thesis by A2,Def7;
  end;
