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 Th43:
  B is non-descending implies Intersection superior_setsequence B = Union B
proof
  assume
A1: B is non-descending;
  now
    let x be object;
    assume
A2: x in Intersection superior_setsequence B;
    now
      let n;
      (superior_setsequence B).n = Union B by A1,Th42;
      hence x in Union B by A2,PROB_1:13;
    end;
    hence x in Union B;
  end;
  then
A3: Intersection superior_setsequence(B) c= Union B;
  now
    let y be object;
    assume y in Union B;
    then for n being Nat holds y in (superior_setsequence(B)).n by A1,Th42;
    hence y in Intersection superior_setsequence(B) by PROB_1:13;
  end;
  then Union B c= Intersection superior_setsequence(B);
  hence thesis by A3,XBOOLE_0:def 10;
end;
