
theorem
  for n be set, L be add-associative right_zeroed right_complementable
  non empty addLoopStr, p be Series of n, L holds p = - -p
   proof
     let n be set,L be add-associative right_zeroed right_complementable
     non empty addLoopStr, p be Series of n, L;
     set r = -p;
A1: dom p = Bags n by FUNCT_2:def 1;
A2: dom -p = dom p by VFUNCT_1:def 5;
A3: dom - -p = dom -p by VFUNCT_1:def 5;
    now
      let x be Element of Bags n;
      assume x in dom p;
A4:   p.x = p/.x;
      thus p.x = - - (p.x) by RLVECT_1:17
      .= - ((-p)/.x) by A1,A2,A4,VFUNCT_1:def 5
      .= (- -p)/.x by A1,A2,A3,VFUNCT_1:def 5
      .= (-r).x;
    end;
    hence thesis by A1;
  end;
