reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

theorem Th48:
  for f be linear-transformation of V1,V2 holds 
  rank f = the_rank_of AutMt(f,b1,b2)
proof
  let f be linear-transformation of V1,V2;
  set A=AutMt(f,b1,b2);
  per cases;
  suppose
A1: len b1=0;
    then len A=0 by MATRIX_0:def 2;
    then dim V1= rank(f) + nullity(f) & the_rank_of A=0 by MATRIX13:74
,RANKNULL:44;
    hence thesis by A1,Th21;
  end;
  suppose
A2: len b1>0& len b2=0;
    then width A=0 by MATRIX_0:23;
    then
A3: the_rank_of A=0 by MATRIX13:74;
    dim V2=0 by A2,Th21;
    hence thesis by A3,VECTSP_9:25;
  end;
  suppose
A4: len b1>0 & len b2>0;
A5: rank f+nullity f = dim V1 by RANKNULL:44
      .= len b1 by Th21;
    nullity f = nullity Mx2Tran(A,b1,b2) by Th34
      .= len b1 - the_rank_of A by A4,Lm7;
    hence thesis by A5;
  end;
end;
