
theorem Th35:
  for K be add-associative right_zeroed right_complementable
Abelian associative well-unital distributive non empty doubleLoopStr for V,W
  be VectSp of K for v,u be Vector of V, w be Vector of W, f be additiveSAF
  homogeneousSAF Form of V,W holds f.(v-u,w) = f.(v,w) - f.(u,w)
proof
  let K be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
  let V,W be VectSp of K, v,w be Vector of V, y be Vector of W;
  let f be additiveSAF homogeneousSAF Form of V,W;
  thus f.(v-w,y) = f.(v+(-w),y) by RLVECT_1:def 11
    .= f.(v,y) +f.(-w,y) by Th26
    .= f.(v,y) +f.((-1.K)* w,y) by VECTSP_1:14
    .= f.(v,y) +(-1.K)*f.(w,y) by Th31
    .= f.(v,y) + -(1.K * f.(w,y)) by VECTSP_1:9
    .= f.(v,y) -(1.K * f.(w,y)) by RLVECT_1:def 11
    .= f.(v,y) - f.( w,y);
end;
