reserve x for set;

theorem Th6:
  for L1, L2 being /\-complete up-complete Semilattice st the
RelStr of L1 = the RelStr of L2 for N1 being NetStr over L1, a being set st [N1
  ,a] in lim_inf-Convergence L1 ex N2 being strict net of L2 st [N2,a] in
  lim_inf-Convergence L2 & the RelStr of N1 = the RelStr of N2 & the mapping of
  N1 = the mapping of N2
proof
  let L1, L2 be /\-complete up-complete Semilattice such that
A1: the RelStr of L1 = the RelStr of L2;
  let N1 be NetStr over L1, a be set;
  assume
A2: [N1,a] in lim_inf-Convergence L1;
  lim_inf-Convergence L1 c= [:NetUniv L1, the carrier of L1:] by
YELLOW_6:def 18;
  then consider N, x being object such that
A3: N in NetUniv L1 and
A4: x in the carrier of L1 and
A5: [N1,a] = [N,x] by A2,ZFMISC_1:def 2;
  reconsider x as Element of L1 by A4;
A6: N = N1 by A5,XTUPLE_0:1;
  then consider N2 being strict net of L2 such that
A7: N2 in NetUniv L2 and
A8: the RelStr of N1 = the RelStr of N2 & the mapping of N1 = the
  mapping of N2 by A1,A3,Th3;
  ex N being strict net of L1 st N = N1 & the carrier of N in
  the_universe_of the carrier of L1 by A3,A6,YELLOW_6:def 11;
  then reconsider N1 as strict net of L1;
A9: now
    let M being subnet of N2;
    consider M1 being strict subnet of N1 such that
A10: the RelStr of M = the RelStr of M1 & the mapping of M = the
    mapping of M1 by A1,A8,Th5;
    thus x = lim_inf M1 by A2,A3,A5,A6,WAYBEL28:def 3
      .= lim_inf M by A1,A10,Th4;
  end;
  take N2;
  x = a by A5,XTUPLE_0:1;
  hence thesis by A1,A7,A8,A9,WAYBEL28:def 3;
end;
