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;
reserve W for non empty RightModStr over K;

theorem Th13:
  for K,L being Ring, V being non empty ModuleStr over K for W
  being non empty RightModStr over L for v1,v2 being Vector of V, w1,w2 being
  Vector of W st W=opp(V) & v1=w1 & v2=w2 holds w1+w2=v1+v2
proof
  let K,L be Ring, V be non empty ModuleStr over K;
A1: the addLoopStr of opp(V) = the addLoopStr of V by Th7;
  let W be non empty RightModStr over L, v1,v2 be Vector of V, w1,w2 be Vector
  of W;
  assume W=opp(V) & v1=w1 & v2=w2;
  hence thesis by A1;
end;
