let X be set ; :: thesis: for p being FinSequence of bool X holds { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } is Subset-Family of X
let p be FinSequence of bool X; :: thesis: { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } is Subset-Family of X
{ (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } c= bool X

proof

let z be set ; :: according to TARSKI:def 3 :: thesis: ( not z in { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } or z in bool X )
assume z in { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } ; :: thesis: z in bool X
then ex q being FinSequence of BOOLEAN st
( z = Intersect (rng (MergeSequence (p,q))) & len q = len p ) ;
hence z in bool X ; :: thesis: verum

end;

hence { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } is Subset-Family of X ; :: thesis: verum