reserve x for set;

theorem
  for L1, L2 being /\-complete up-complete Semilattice st the RelStr of
  L1 = the RelStr of L2 holds ConvergenceSpace lim_inf-Convergence L1 =
  ConvergenceSpace lim_inf-Convergence L2
proof
  let L1, L2 be /\-complete up-complete Semilattice such that
A1: the RelStr of L1 = the RelStr of L2;
  set C2 = lim_inf-Convergence L2;
  set C1 = lim_inf-Convergence L1;
  set T1 = ConvergenceSpace C1;
  set T2 = ConvergenceSpace C2;
  the topology of T1 c= the topology of T2 & the topology of T2 c= the
  topology of T1 by A1,Lm3;
  then the carrier of T2 = the carrier of L2 & the topology of T1 = the
  topology of T2 by YELLOW_6:def 24;
  hence thesis by A1,YELLOW_6:def 24;
end;
