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 Th24:
  v1 in UnionKers f iff ex n st (f|^n).v1 = 0.V1
proof
  hereby
    assume v1 in UnionKers f;
    then v1 in the carrier of UnionKers f;
    then v1 in {w where w is Vector of V1:ex n st (f|^n).w = 0.V1} by Def5;
    then ex w be Vector of V1 st v1=w & ex m st (f|^m).w=0.V1;
    hence ex n st (f|^n).v1=0.V1;
  end;
  assume ex n st (f|^n).v1=0.V1;
  then v1 in {w where w is Vector of V1:ex n st (f|^n).w = 0.V1};
  then v1 in the carrier of UnionKers f by Def5;
  hence thesis;
end;
