
theorem Th33:
  for m being Nat, L being well-unital non empty doubleLoopStr,
  x being Element of L holds DFT(0_.(L),x,m) = 0_.(L)
proof
  let m be Nat, L be well-unital non empty doubleLoopStr, x be Element of L;
  set q = DFT(0_.(L),x,m);
A1: now
    let u be object;
    assume u in dom q;
    then reconsider n = u as Element of NAT by FUNCT_2:def 1;
    per cases;
    suppose
      n < m;
      hence q.u = eval(0_.(L),x |^ n) by Def6
        .= 0.L by POLYNOM4:17
        .= (0_.(L)).n by FUNCOP_1:7
        .= (0_.(L)).u;
    end;
    suppose
      n >= m;
      hence q.u = 0.L by Def6
        .= (0_.(L)).n by FUNCOP_1:7
        .= (0_.(L)).u;
    end;
  end;
  dom q = NAT by FUNCT_2:def 1
    .= dom 0_.(L) by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
