now :: thesis: for n being Nat holds (superior_setsequence S) . n in Si

let n be Nat; :: thesis: (superior_setsequence S) . n in Si
reconsider DSeq = S ^\ n as SetSequence of X ;
reconsider n1 = n as Nat ;
A1: Union DSeq in Si by PROB_1:17;
Union DSeq = union (rng DSeq) by CARD_3:def 4;
then Union DSeq = union { (S . k) where k is Nat : n1 <= k } by Th6;
hence (superior_setsequence S) . n in Si by A1, Def3; :: thesis: verum

end;

then rng (superior_setsequence S) c= Si by NAT_1:52;
hence superior_setsequence S is SetSequence of Si by RELAT_1:def 19; :: thesis: verum