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;
