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 Th53:
  B is non-ascending implies Union inferior_setsequence(B) = Intersection B
proof
  assume
A1: B is non-ascending;
  now
    let x be object;
    assume x in Union inferior_setsequence(B);
    then ex k being Nat st x in (inferior_setsequence(B)).k by PROB_1:12;
    hence x in Intersection B by A1,Th52;
  end;
  then
A2: Union inferior_setsequence(B) c= Intersection B;
  now
    let y be object;
    assume y in Intersection B;
    then y in (inferior_setsequence(B)).0 by Th17;
    hence y in Union inferior_setsequence(B) by PROB_1:12;
  end;
  then Intersection B c= Union inferior_setsequence(B);
  hence thesis by A2,XBOOLE_0:def 10;
end;
