reserve A,B,C for non empty set,
  f for Function of [:A,B:],C;
reserve K for non empty doubleLoopStr;

theorem
  the addLoopStr of opp(K) = the addLoopStr of K & (K is add-associative
  right_zeroed right_complementable implies comp opp K = comp K) & for x being
  set holds x is Scalar of opp(K) iff x is Scalar of K
proof
  thus the addLoopStr of opp(K) = the addLoopStr of K;
  hereby
    assume
A1: K is add-associative right_zeroed right_complementable;
A2: for x be object st x in the carrier of K
      holds (comp opp K).x = (comp K). x
    proof
      let x be object;
      assume x in the carrier of K;
      then reconsider y = x as Element of K;
      reconsider z = y as Element of opp K;
A3:   -y = -z by A1,Lm3;
      thus (comp opp K).x = -z by VECTSP_1:def 13
        .= (comp K).x by A3,VECTSP_1:def 13;
    end;
    dom comp opp K = the carrier of K & dom comp K = the carrier of K by
FUNCT_2:def 1;
    hence comp opp K = comp K by A2,FUNCT_1:2;
  end;
  let x be set;
  thus thesis;
end;
