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 scalar of V = [: the carrier of V, the carrier of V:] by FUNCT_2:def 1;
( dom the addF of V = [: the carrier of V, the carrier of V:] & dom the Mult of V = [:REAL, 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, RELAT_1:69; :: thesis: verum