theorem
  for f be linear-transformation of V1,V2 st dim V1=dim V2 holds 
  ker f is non trivial iff Det AutEqMt(f,b1,b2) = 0.K
proof
  let f be linear-transformation of V1,V2 such that
A1: dim V1=dim V2;
  set A=AutEqMt(f,b1,b2);
  dim V2=len b2 by Th21;
  then
A2: A=AutMt(f,b1,b2) by A1,Def2,Th21;
A3: dim V1=rank f+nullity f by RANKNULL:44;
A4: len b1=dim V1 & rank f=the_rank_of AutMt(f,b1,b2) by Th21,Th48;
  hereby
    assume ker f is non trivial;
    then rank f <> dim V1 by A3,Th42;
    hence Det A = 0.K by A4,A2,MATRIX13:83;
  end;
  assume Det A=0.K;
  then dim ker f<>0 by A4,A2,A3,MATRIX13:83;
  hence thesis by Th42;
end;
