Additive structures with a total defined scalar multiplication from the left by a commutative semi ring. The entities of v are called vector.

Properties Let v b a Vectorial structure, then holds:

1. For all s in Scalar v and v in v holds: s!v is valid and root (s!v) == root v.
2. For all v in v holds: 0!v == zero (root v).
3. For all s in Scalar v and r in Root v holds s!zero r == zero r.
4. For all r, s in Scalar v and v in v holds: (r + s)!v == r!v + s!v.
5. For all s in Scalar v and v, w in v with root v == root w holds: s!(v + w) == s!v + s!w.
6. For all v in v holds: 1!v == v.
7. For all r, s in Scalar v and v in v holds: (r*s)!v == r!(s!v).

type representing the class of k-Vectorial structures.

helper class to avoid undecidable instances.

list of scalars and vectors, having all the same given root.

Property Let VectorSheaf r svs be in VectorSheaf v for a Vectorial-structure v, then holds: root v == r, for all (_,v) in svs.

Vectorial structures with a partially defined scalar product.

Properties

1. For all v, w holds: if root v == root w then v <!> w is valid, otherwise a UndefinedScalarproduct-exception will be thrown.
2. For all u holds: u <!> zero (root u) == rZero.
3. For all u, v and w with root u == root w and root w == root v holds: u <!> (v + w) == u <!> v + u <!> w.
4. For all w holds: zero (root w) <!> w == rZero.
5. For all u, v and w with root w == root u and root u == root v holds: (u + v) <!> w == u <!> w + v' !' w.