let Y0 be non empty SubSpace of Y; :: thesis: ( Y0 is maximal_discrete implies Y0 is discrete )
reconsider A = the carrier of Y0 as Subset of Y by Lm1;
assume Y0 is maximal_discrete ; :: thesis: Y0 is discrete
then A is maximal_discrete ;
then A is discrete ;
hence Y0 is discrete by Th20; :: thesis: verum