reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a for Element of K;
reserve V for non trivial VectSp of K,
  V1,V2 for VectSp of K,
  f for linear-transformation of V1,V1,
  v,w for Vector of V,
  v1 for Vector of V1,
  L for Scalar of K;
reserve S for 1-sorted,
  F for Function of S,S;

theorem Th34:
  UnionKers f = ker (f|^n) implies ker (f|^n) /\ im (f|^n) = (0). V1
proof
  set KER=ker (f|^n);
  set IM = im (f|^n);
  assume
A1: UnionKers f = ker (f|^n);
  the carrier of KER /\ IM c= {0.V1}
  proof
    let x be object;
    assume x in the carrier of KER/\IM;
    then
A2: x in KER/\IM;
    then x in V1 by VECTSP_4:9;
    then reconsider v=x as Vector of V1;
    v in IM by A2,VECTSP_5:3;
    then consider w be Element of V1 such that
A3: (f|^n).w=v by RANKNULL:13;
A4: dom (f|^n) = the carrier of V1 by FUNCT_2:def 1;
    v in KER by A2,VECTSP_5:3;
    then 0.V1 = (f|^n).((f|^n).w) by A3,RANKNULL:10
      .= ((f|^n)*(f|^n)).w by A4,FUNCT_1:13
      .= (f|^(n+n)).w by Th20;
    then w in ker(f|^n) by A1,Th24;
    then v=0.V1 by A3,RANKNULL:10;
    hence thesis by TARSKI:def 1;
  end;
  then the carrier of KER/\IM = {0.V1} by ZFMISC_1:33
    .= the carrier of (0).V1 by VECTSP_4:def 3;
  hence thesis by VECTSP_4:29;
end;
