take V ; :: thesis: ( the carrier of V c= the carrier of V & 0. V = 0. V & the addF of V = the addF of V | [: the carrier of V, the carrier of V:] & the rmult of V = the rmult of V | [: the carrier of V, the carrier of R:] )
thus ( the carrier of V c= the carrier of V & 0. V = 0. V ) ; :: thesis: ( the addF of V = the addF of V | [: the carrier of V, the carrier of V:] & the rmult of V = the rmult of V | [: the carrier of V, the carrier of R:] )
thus ( the addF of V = the addF of V | [: the carrier of V, the carrier of V:] & the rmult of V = the rmult of V | [: the carrier of V, the carrier of R:] ) by RELSET_1:19; :: thesis: verum