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 lmult of V = the lmult of V | [: the carrier of GF, the carrier of V:] )

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 lmult of V = the lmult of V | [: the carrier of GF, the carrier of V:] )

thus ( the addF of V = the addF of V || the carrier of V & the lmult of V = the lmult of V | [: the carrier of GF, the carrier of V:] ) by RELSET_1:19; :: thesis: verum

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 lmult of V = the lmult of V | [: the carrier of GF, the carrier of V:] )

thus ( the addF of V = the addF of V || the carrier of V & the lmult of V = the lmult of V | [: the carrier of GF, the carrier of V:] ) by RELSET_1:19; :: thesis: verum