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 Mult of V = the Mult of V | [:REAL ,the carrier of V:] & the scalar of V = the scalar of V || the carrier of V )
A1: dom the addF of V = [:the carrier of V,the carrier of V:] by FUNCT_2:def 1;
A2: dom the Mult of V = [:REAL ,the carrier of V:] by FUNCT_2:def 1;
dom the scalar of V = [:the carrier of V,the carrier of V:] by FUNCT_2:def 1;
hence ( 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 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 A1, A2, RELAT_1:98; :: thesis: verum