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

theorem Th15:
  for K be doubleLoopStr for V be non empty ModuleStr over K for f
  be nilpotent Function of V,V holds deg f = 0 iff [#]V = {0.V}
proof
  let K be doubleLoopStr;
  let V be non empty ModuleStr over K;
  let f be nilpotent Function of V,V;
  hereby
    assume
A1: deg f=0;
    [#]V c= {0.V}
    proof
      let x be object such that
A2:   x in [#]V;
      id V = f|^0 by VECTSP11:18
        .= ZeroMap(V,V) by A1,Def5
        .= (the carrier of V)-->0.V by GRCAT_1:def 7;
      then x = ((the carrier of V)-->0.V).x by A2,FUNCT_1:18
        .= 0.V by A2,FUNCOP_1:7;
      hence thesis by TARSKI:def 1;
    end;
    hence [#]V ={0.V} by ZFMISC_1:33;
  end;
  assume
A3: [#]V={0.V};
  now
    let x be object;
    assume x in dom (f|^0);
    then reconsider v=x as Vector of V by FUNCT_2:def 1;
    thus (f|^0).x = (id V).v by VECTSP11:18
      .= 0.V by A3,TARSKI:def 1;
  end;
  then (f|^0) = (dom (f|^0))-->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 thesis by Def5;
end;
