theorem Th9:
  the addLoopStr of opp(W) = the addLoopStr of W & for x being set
  holds x is Vector of W iff x is Vector of opp(W)
proof
  reconsider p = ~(the rmult of W) as Function of [:the carrier of opp(K), the
  carrier of W:], the carrier of W;
A1: opp(W) = ModuleStr (# the carrier of W, the addF of W, 0.W, p #) by Def4;
  hence the addLoopStr of opp(W) = the addLoopStr of W;
  let x be set;
  thus thesis by A1;
end;
