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
  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;
