let V be RealUnitarySpace; :: thesis: V is Subspace of V
thus ( the carrier of V c= the carrier of V & 0. V = 0. V ) ; :: according to RUSUB_1:def 1 :: thesis: ( the addF of V = the addF of V || the carrier of V & the Mult of V = the Mult of V | [:REAL, the carrier of V:] & the scalar of V = the scalar of V || the carrier of V )
thus ( the addF of V = the addF of V || the carrier of V & the Mult of V = the Mult of V | [:REAL, the carrier of V:] & the scalar of V = the scalar of V || the carrier of V ) by RELSET_1:19; :: thesis: verum