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 Th28:
  f|ker (f|^n) is linear-transformation of ker (f|^n),ker (f|^n)
proof
  set KER=ker (f|^n);
  rng (f|KER)c= the carrier of KER
  proof
    let y be object;
    assume y in rng (f|KER);
    then consider x being object such that
A1: x in dom (f|KER) and
A2: (f|KER).x=y by FUNCT_1:def 3;
    x in the carrier of KER by A1,FUNCT_2:def 1;
    then
A3: x in KER;
    then x in V1 by VECTSP_4:9;
    then reconsider v=x as Vector of V1;
A4: dom f = the carrier of V1 by FUNCT_2:def 1;
    (f|^n).v = 0.V1 by A3,RANKNULL:10;
    then 0.V1 = (f|^(n+1)).v by Th23
      .= ((f|^n)*(f|^1)).v by Th20
      .= ((f|^n)*f).v by Th19
      .= (f|^n).(f.v) by A4,FUNCT_1:13;
    then
A5: f.v in KER by RANKNULL:10;
    y=f.v by A1,A2,FUNCT_1:47;
    hence thesis by A5;
  end;
  hence thesis by Lm1,FUNCT_2:6;
end;
