deffunc H₁( set ) -> set = bool (X . n);
consider p being FinSequence such that
A1: len p = n and
A2: for i being Nat st i in dom p holds
p . i = H₁(i) from FINSEQ_1:sch 2();
reconsider p = p as n -element FinSequence by A1, CARD_1:def 7;
take p ; :: thesis:

let i be Nat; :: thesis:
assume A3: i in Seg n ; :: thesis:
reconsider Xi = X . i as set ;
A4: bool Xi is semiring_of_sets of Xi by SRINGS_2:5;
i in dom p by A3, A1, FINSEQ_1:def 3;
hence p . i is semiring_of_sets of (X . i) by A2, A4; :: thesis:

end;

then reconsider p = p as SemiringFamily of X by Def2;

let i be Nat; :: thesis:
assume i in Seg n ; :: thesis:
then i in dom p by A1, FINSEQ_1:def 3;
then p . i = bool (X . i) by A2;
hence p . i is cap-closed ; :: thesis:

end;

then p is cap-closed-yielding ;
hence ( p is SemiringFamily of X & p is cap-closed-yielding ) ; :: thesis: