let Y0 be non empty SubSpace of Y; :: thesis: ( Y0 is maximal_T_0 implies Y0 is V56() )
reconsider A = the carrier of Y0 as Subset of Y by Lm1;
assume Y0 is maximal_T_0 ; :: thesis: Y0 is V56()
then A is maximal_T_0 by Th11;
then A is T_0 by Def4;
hence Y0 is V56() by TSP_1:13; :: thesis: verum