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 Th15:
  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 V=opp(W) & 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: V=opp(W) & x=y & v=w;
  set o = the rmult of W;
A2: opp(o) = the lmult of opp(W) by Th10;
  thus w*y = o.(w,y) by VECTSP_2:def 7
    .= x*v by A1,A2,FUNCT_4:def 8;
end;
