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 Th44:
  B is non-descending implies B.n c= (inferior_setsequence B).(n+1 )
proof
  set Y = {B.k : n+1 <= k};
  assume
A1: B is non-descending;
  let x be object;
  assume
A2: x in B.n;
A3: now
    let y be set;
    assume y in Y;
    then consider k1 being Nat such that
A4: y=B.k1 and
A5: n+1 <= k1;
    n <= k1 by A5,NAT_1:13;
    then B.n c= B.k1 by A1,PROB_1:def 5;
    hence x in y by A2,A4;
  end;
A6: Y <> {} by Th1;
  (inferior_setsequence B).(n+1) = meet {B.k : n+1 <= k} by Def2;
  hence thesis by A6,A3,SETFAM_1:def 1;
end;
