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 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 being 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;
