consider p being FinSequence of bool X such that
len p = card Y and
rng p = Y and
A1: Components Y = { (Intersect (rng (MergeSequence (p,q)))) where q is FinSequence of BOOLEAN : len q = len p } by Def2;
deffunc H₁( Element of BOOLEAN * ) -> Element of bool X = Intersect (rng (MergeSequence (p,X)));
A2: Components Y c= { (Intersect (rng (MergeSequence (p,q)))) where q is Element of BOOLEAN * : q in (len p) -tuples_on BOOLEAN }

proof

let z be object ; :: according to TARSKI:def 3 :: thesis:
assume z in Components Y ; :: thesis:
then consider q being FinSequence of BOOLEAN such that
A3: ( z = Intersect (rng (MergeSequence (p,q))) & len q = len p ) by A1;
( q is Element of BOOLEAN * & q is Element of (len q) -tuples_on BOOLEAN ) by FINSEQ_1:def 11, FINSEQ_2:92;
hence z in { (Intersect (rng (MergeSequence (p,q)))) where q is Element of BOOLEAN * : q in (len p) -tuples_on BOOLEAN } by A3; :: thesis:

end;

A4: (len p) -tuples_on BOOLEAN is finite ;
{ H₁(q) where q is Element of BOOLEAN * : q in (len p) -tuples_on BOOLEAN } is finite from FRAENKEL:sch 21(A4);
hence Components Y is finite by A2; :: thesis: