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

theorem Th7:
  the addLoopStr of opp(V) = the addLoopStr of V & for x being set
  holds x is Vector of V iff x is Vector of opp(V)
proof
  reconsider p = ~(the lmult of V) as Function of [:the carrier of V, the
  carrier of opp(K):], the carrier of V;
A1: opp(V) = RightModStr (# the carrier of V, the addF of V, 0.V, p #) by Def2;
  hence the addLoopStr of opp(V) = the addLoopStr of V;
  let x be set;
  thus thesis by A1;
end;
