theorem Th63:
  B is non-descending implies B is convergent & lim B = Union B
proof
  assume
A1: B is non-descending;
  then lim_sup B = Union B & lim_inf B = Union B by Th43,Th46;
  hence B is convergent by KURATO_0:def 5;
  thus thesis by A1,Th43;
end;
