
theorem
for L being add-associative right_zeroed right_complementable
            left-distributive non empty doubleLoopStr,
    p being sequence of L
holds 0.L * p = 0_.(L)
proof
let L be add-associative right_zeroed right_complementable
         left-distributive non empty doubleLoopStr,
    p be sequence of L;
set t = 0.L * p;
now let x be object;
  assume x in NAT;
  then reconsider i = x as Element of NAT;
  thus t.x = 0.L * (p.i) by POLYNOM5:def 4
          .= (0_.(L)).x by FUNCOP_1:7;
  end;
hence thesis by FUNCT_2:12;
end;
