
theorem Th16:
  for X being set, L being add-associative right_zeroed
right_complementable non empty doubleLoopStr, a being Element of L holds -(a
  |(X,L)) = (-a) |(X,L)
proof
  let n be set, L be add-associative right_zeroed right_complementable non
  empty doubleLoopStr, a be Element of L;
A1: now
    let u be object;
    assume u in dom((-a) |(n,L));
    then reconsider u9 = u as Element of Bags n;
    now
      per cases;
      case
A2:     u9 = EmptyBag n;
        hence -((a |(n,L)).u9) = - a by POLYNOM7:18
          .= ((-a) |(n,L)).u9 by A2,POLYNOM7:18;
      end;
      case
A3:     u9 <> EmptyBag n;
        -0.L = 0.L by RLVECT_1:12;
        hence -((a |(n,L)).u9) = 0.L by A3,POLYNOM7:18
          .= ((-a) |(n,L)).u9 by A3,POLYNOM7:18;
      end;
    end;
    hence ((-a) |(n,L)).u = (- (a |(n,L))).u by POLYNOM1:17;
  end;
  dom(- (a |(n,L))) = Bags n by FUNCT_2:def 1
    .= dom((-a) |(n,L)) by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
