reserve n,m,k,k1,k2,i,j for Nat;
reserve x,y,z for object,X,Y,Z for set;
reserve A for Subset of X;
reserve B,A1,A2,A3 for SetSequence of X;
reserve Si for SigmaField of X;
reserve S,S1,S2,S3 for SetSequence of Si;

theorem Th21:
 for n being Nat holds
  (inferior_setsequence B).n = (inferior_setsequence B).(n+1) /\ B .n
proof
    let n be Nat;
A1: (inferior_setsequence B).n = meet {B.k1 : n <= k1} by Def2;
A2: {B.k1: n <= k1} = {B.k2 : n+1 <=k2} \/ {B.n} by Th2;
A3: (inferior_setsequence B).(n+1) = meet {B.k2 : n+1 <= k2} by Def2;
A4: {B.k1 : n <= k1} <> {} by Th1;
A5: now
      let x be object;
      assume that
A6:   x in (inferior_setsequence B).(n+1) and
A7:   x in B.n;
      for Z st Z in {B.k2 : n <= k2} holds x in Z
      proof
        let Z;
        assume Z in {B.k1 : n <= k1};
        then consider Z1 being set such that
A8:     Z=Z1 & Z1 in {B.k1 : n <= k1};
        consider k11 being Nat such that
A9:     Z1=B.k11 & n <= k11 by A8;
        now
          per cases by A9,Lm1;
          suppose
            Z1=B.k11 & n = k11;
            hence x in Z1 by A7;
          end;
          suppose
            Z1=B.k11 & n+1 <= k11;
            then Z1 in {B.k2 : n+1 <= k2};
            hence x in Z1 by A3,A6,SETFAM_1:def 1;
          end;
        end;
        hence thesis by A8;
      end;
      then x in meet {B.k2 : n <= k2} by A4,SETFAM_1:def 1;
      hence x in (inferior_setsequence B).n by Def2;
    end;
    {B.k2 : n+1 <= k2} <> {} by Th1;
    then
A10: (inferior_setsequence B).(n) c= (inferior_setsequence B).(n+1) by A1,A3,A2
,SETFAM_1:6,XBOOLE_1:7;
    now
      let x be object;
A11:  B.n in {B.k1 : n <= k1};
      assume x in (inferior_setsequence B).n;
      hence x in (inferior_setsequence B).(n+1) & x in B.n by A1,A10,A11,
SETFAM_1:def 1;
    end;
    hence
    (inferior_setsequence B).n = (inferior_setsequence B).(n+1) /\ B.n by A5,
XBOOLE_0:def 4;
end;
