theorem
  IB <> (.B.> & IB is prime iff IB is max-ideal
proof
  reconsider F = IB.: as Filter of B.:;
  <.B.:.) = (.B.>;
  then F <> (.B.> & F is prime iff F is being_ultrafilter by FILTER_0:45;
  hence thesis by Th32,Th43;
end;
