reserve G for non empty addLoopStr;
reserve x for Element of G;
reserve M for non empty MidStr;
reserve p,q,r for Point of M;
reserve w for Function of [:the carrier of M,the carrier of M:], the carrier
  of G;
reserve S for non empty set;
reserve a,b,b9,c,c9,d for Element of S;
reserve w for Function of [:S,S:],the carrier of G;
reserve G for add-associative right_zeroed right_complementable non empty
  addLoopStr;
reserve x for Element of G;
reserve w for Function of [:S,S:],the carrier of G;
reserve G for add-associative right_zeroed right_complementable Abelian non
  empty addLoopStr;
reserve x for Element of G;

theorem Th12:
  Double (-x) = -Double x
proof
  0.G = Double 0.G by RLVECT_1:def 4
    .= Double (x+-x) by RLVECT_1:def 10
    .= Double x + Double (-x) by Th10;
  hence thesis by RLVECT_1:6;
end;
