set F = { A where A is Subset of V : A is linearly-independent } ;
set X = the carrier of V;
{ A where A is Subset of V : A is linearly-independent } c= bool the carrier of V

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { A where A is Subset of V : A is linearly-independent } or x in bool the carrier of V )
assume x in { A where A is Subset of V : A is linearly-independent } ; :: thesis: x in bool the carrier of V
then ex A being Subset of the carrier of V st
( x = A & A is linearly-independent ) ;
hence x in bool the carrier of V ; :: thesis: verum

end;

then reconsider F = { A where A is Subset of V : A is linearly-independent } as Subset-Family of the carrier of V ;
take M = TopStruct(# the carrier of V,F #); :: thesis: ( the carrier of M = the carrier of V & the_family_of M = { A where A is Subset of V : A is linearly-independent } )
thus ( the carrier of M = the carrier of V & the_family_of M = { A where A is Subset of V : A is linearly-independent } ) ; :: thesis: verum