theorem Th80:
  S is non-descending implies S is convergent & lim S = Union S
proof
  assume
A1: S is non-descending;
  then lim_sup S = Union S & lim_inf S = Union S by Th78,Th79;
  hence S is convergent by KURATO_0:def 5;
  thus thesis by A1,Th78;
end;
