reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,L for Element of K;

theorem Th13:
  for K be doubleLoopStr for V be non empty ModuleStr over K for f
  be Function of V,V holds f is nilpotent iff ex n st f|^n = ZeroMap(V,V)
proof
  let K be doubleLoopStr;
  let V be non empty ModuleStr over K;
  let f be Function of V,V;
  hereby
    assume f is nilpotent;
    then consider n such that
A1: for v be Vector of V holds (f|^n).v=0.V;
    now
      let x be object;
      assume x in dom (f|^n);
      then reconsider v=x as Vector of V by FUNCT_2:def 1;
      thus (f|^n).x = (f|^n).v .= 0.V by A1;
    end;
    then (f|^n) = (dom (f|^n))-->0.V by FUNCOP_1:11
      .= (the carrier of V)-->0.V by FUNCT_2:def 1
      .= ZeroMap(V,V) by GRCAT_1:def 7;
    hence ex n st f|^n = ZeroMap(V,V);
  end;
  given n such that
A2: f|^n = ZeroMap(V,V);
  take n;
  let v be Vector of V;
  thus (f|^n).v = ((the carrier of V)-->0.V).v by A2,GRCAT_1:def 7
    .= 0.V;
end;
