let L1, L2 be up-complete /\-complete Semilattice; :: thesis: ( RelStr(# the carrier of L1, the InternalRel of L1 #) = RelStr(# the carrier of L2, the InternalRel of L2 #) implies for N1 being net of L1
for N2 being net of L2 st RelStr(# the carrier of N1, the InternalRel of N1 #) = RelStr(# the carrier of N2, the InternalRel of N2 #) & the mapping of N1 = the mapping of N2 holds
lim_inf N1 = lim_inf N2 )
assume A1: RelStr(# the carrier of L1, the InternalRel of L1 #) = RelStr(# the carrier of L2, the InternalRel of L2 #) ; :: thesis: for N1 being net of L1
for N2 being net of L2 st RelStr(# the carrier of N1, the InternalRel of N1 #) = RelStr(# the carrier of N2, the InternalRel of N2 #) & the mapping of N1 = the mapping of N2 holds
lim_inf N1 = lim_inf N2
let N1 be net of L1; :: thesis: for N2 being net of L2 st RelStr(# the carrier of N1, the InternalRel of N1 #) = RelStr(# the carrier of N2, the InternalRel of N2 #) & the mapping of N1 = the mapping of N2 holds
lim_inf N1 = lim_inf N2
let N2 be net of L2; :: thesis: ( RelStr(# the carrier of N1, the InternalRel of N1 #) = RelStr(# the carrier of N2, the InternalRel of N2 #) & the mapping of N1 = the mapping of N2 implies lim_inf N1 = lim_inf N2 )
assume A2: ( RelStr(# the carrier of N1, the InternalRel of N1 #) = RelStr(# the carrier of N2, the InternalRel of N2 #) & the mapping of N1 = the mapping of N2 ) ; :: thesis: lim_inf N1 = lim_inf N2
deffunc H1( Element of N2) -> set = { (N2 . i) where i is Element of N2 : i >= $1 } ;
deffunc H2( Element of N1) -> set = { (N1 . i) where i is Element of N1 : i >= $1 } ;
set U1 = { ("/\" (H2(j),L1)) where j is Element of N1 : verum } ;
set U2 = { ("/\" (H1(j),L2)) where j is Element of N2 : verum } ;
A3: ( lim_inf N1 = "\/" ( { ("/\" (H2(j),L1)) where j is Element of N1 : verum } ,L1) & lim_inf N2 = "\/" ( { ("/\" (H1(j),L2)) where j is Element of N2 : verum } ,L2) ) by WAYBEL11:def 6;
{ ("/\" (H2(j),L1)) where j is Element of N1 : verum } c= the carrier of L1
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { ("/\" (H2(j),L1)) where j is Element of N1 : verum } or x in the carrier of L1 )
assume x in { ("/\" (H2(j),L1)) where j is Element of N1 : verum } ; :: thesis: x in the carrier of L1
then ex j being Element of N1 st x = "/\" (H2(j),L1) ;
hence x in the carrier of L1 ; :: thesis: verum
end;
then reconsider U1 = { ("/\" (H2(j),L1)) where j is Element of N1 : verum } as Subset of L1 ;
( not U1 is empty & U1 is directed ) by WAYBEL32:7;
then A4: ex_sup_of U1,L1 by WAYBEL_0:75;
( U1 c= { ("/\" (H1(j),L2)) where j is Element of N2 : verum } & { ("/\" (H1(j),L2)) where j is Element of N2 : verum } c= U1 ) by A1, A2, Lm2;
then U1 = { ("/\" (H1(j),L2)) where j is Element of N2 : verum } ;
hence lim_inf N1 = lim_inf N2 by A1, A3, A4, YELLOW_0:26; :: thesis: verum