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

let n be Nat; :: thesis: (inferior_setsequence S) . n in Si
reconsider DSeq = S ^\ n as SetSequence of X ;
reconsider n1 = n as Nat ;
for m being Nat holds DSeq . m in Si

proof

let m be Nat; :: thesis: DSeq . m in Si
( DSeq . m = S . (m + n1) & S . (m + n1) in rng S ) by NAT_1:51, NAT_1:def 3;
hence DSeq . m in Si ; :: thesis: verum

end;

then rng DSeq c= Si by NAT_1:52;
then A1: Intersection DSeq in Si by PROB_1:def 6;
Intersection DSeq = meet (rng DSeq) by Th8;
then Intersection DSeq = meet { (S . k) where k is Nat : n1 <= k } by Th6;
hence (inferior_setsequence S) . n in Si by A1, Def2; :: thesis: verum

end;

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