let N be complete Lawson TopLattice; :: thesis: for S being Scott TopAugmentation of N
for x being Element of N holds { (inf A) where A is Subset of S : ( x in A & A in sigma S ) } c= { (inf J) where J is Subset of N : ( x in J & J in lambda N ) }
let S be Scott TopAugmentation of N; :: thesis: for x being Element of N holds { (inf A) where A is Subset of S : ( x in A & A in sigma S ) } c= { (inf J) where J is Subset of N : ( x in J & J in lambda N ) }
let x be Element of N; :: thesis: { (inf A) where A is Subset of S : ( x in A & A in sigma S ) } c= { (inf J) where J is Subset of N : ( x in J & J in lambda N ) }
A1: RelStr(# the carrier of N,the InternalRel of N #) = RelStr(# the carrier of S,the InternalRel of S #) by YELLOW_9:def 4;
set s = { (inf A) where A is Subset of S : ( x in A & A in sigma S ) } ;
set l = { (inf J) where J is Subset of N : ( x in J & J in lambda N ) } ;
let k be set ; :: according to TARSKI:def 3 :: thesis: ( not k in { (inf A) where A is Subset of S : ( x in A & A in sigma S ) } or k in { (inf J) where J is Subset of N : ( x in J & J in lambda N ) } )
assume k in { (inf A) where A is Subset of S : ( x in A & A in sigma S ) } ; :: thesis: k in { (inf J) where J is Subset of N : ( x in J & J in lambda N ) }
then consider J being Subset of S such that
A2: ( k = inf J & x in J & J in sigma S ) ;
reconsider A = J as Subset of N by A1;
ex_inf_of A,N by YELLOW_0:17;
then A3: inf A = inf J by A1, YELLOW_0:27;
sigma N c= lambda N by Th10;
then sigma S c= lambda N by A1, YELLOW_9:52;
hence k in { (inf J) where J is Subset of N : ( x in J & J in lambda N ) } by A2, A3; :: thesis: verum