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 Th12:
  for K,L being Ring, V being non empty ModuleStr over K for W
being non empty RightModStr over L for x being Scalar of K, y being Scalar of L
  , v being Vector of V, w being Vector of W st L=opp(K) & W=opp(V) & x=y & v=w
  holds w*y = x*v
proof
  let K,L be Ring, V be non empty ModuleStr over K;
  let W be non empty RightModStr over L;
  let x be Scalar of K, y be Scalar of L, v be Vector of V, w be Vector of W
  such that
A1: L=opp(K) & W=opp(V) and
A2: x=y & v=w;
  set o = the lmult of V;
  opp(o) = the rmult of opp(V) by Th8;
  hence w*y = (opp(o)).(w,y) by A1,VECTSP_2:def 7
    .= x*v by A2,FUNCT_4:def 8;
end;
