set X = { A where A is Subset of AS : A is being_plane } ;
{ A where A is Subset of AS : A is being_plane } c= bool the carrier of AS

proof

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

end;

hence { A where A is Subset of AS : A is being_plane } is Subset-Family of AS ; :: thesis: verum