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 Th83:
  S is non-ascending implies S is convergent & lim S = Intersection S
proof
  assume
A1: S is non-ascending;
  then lim_sup S = Intersection S & lim_inf S = Intersection S by Th81,Th82;
  hence S is convergent by KURATO_0:def 5;
  thus thesis by A1,Th81;
end;
