
theorem Th36:
  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 be Vector of V, w,t be Vector of W, f be additiveFAF
  homogeneousFAF Form of V,W holds f.(v,w-t) = f.(v,w) - f.(v,t)
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 be Vector of V, y,z be Vector of W, f be
  additiveFAF homogeneousFAF Form of V,W;
  thus f.(v,y-z) = f.(v, y+(-z)) by RLVECT_1:def 11
    .= f.(v,y) + f.(v,-z) by Th27
    .= f.(v,y) + f.(v,(-1.K)* z) by VECTSP_1:14
    .= f.(v,y) + (-1.K)*f.(v,z) by Th32
    .= f.(v,y) + -(1.K * f.(v,z)) by VECTSP_1:9
    .= f.(v,y) -(1.K * f.(v,z)) by RLVECT_1:def 11
    .= f.(v,y) - f.(v,z);
end;
