deffunc H₁( set ) -> Element of bool (bool (X . $1)) = bool (X . $1);
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: for i being Nat st i in Seg n holds
p . i is semialgebra_of_sets of X . i

hereby :: thesis: verum

let i be Nat; :: thesis: ( i in Seg n implies p . i is semialgebra_of_sets of X . i )
assume A3: i in Seg n ; :: thesis: p . i is semialgebra_of_sets of X . i
reconsider Xi = X . i as set ;
reconsider BXi = bool Xi as Subset-Family of Xi ;
A4: BXi is with_empty_element cap-finite-partition-closed diff-c=-finite-partition-closed with_countable_Cover Subset-Family of Xi ;
A5: i in dom p by A3, A1, FINSEQ_1:def 3;
then A6: p . i is with_empty_element cap-closed semi-diff-closed Subset-Family of (X . i) by A2, A4;
X . i in bool (X . i) by ZFMISC_1:def 1;
then X . i in p . i by A2, A5;
hence p . i is semialgebra_of_sets of X . i by A6, SRINGS_3:def 6; :: thesis: verum

end;