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;
