let N be complete Lawson meet-continuous TopLattice; :: thesis: ( N is continuous iff for x being Element of N holds x = "\/" ( { (inf V) where V is Subset of N : ( x in V & V in lambda N ) } ,N) )
set S = the complete Scott TopAugmentation of N;
A1: RelStr(# the carrier of the complete Scott TopAugmentation of N, the InternalRel of the complete Scott TopAugmentation of N #) = RelStr(# the carrier of N, the InternalRel of N #) by YELLOW_9:def 4;
hereby :: thesis: ( ( for x being Element of N holds x = "\/" ( { (inf V) where V is Subset of N : ( x in V & V in lambda N ) } ,N) ) implies N is continuous )
assume A2: N is continuous ; :: thesis: for x being Element of N holds x = "\/" ( { (inf V) where V is Subset of N : ( x in V & V in lambda N ) } ,N)
then A3: for x being Element of the complete Scott TopAugmentation of N ex J being Basis of x st
for Y being Subset of the complete Scott TopAugmentation of N st Y in J holds
( Y is open & Y is filtered ) by WAYBEL14:35;
let x be Element of N; :: thesis: x = "\/" ( { (inf V) where V is Subset of N : ( x in V & V in lambda N ) } ,N)
InclPoset () is continuous by ;
hence x = "\/" ( { (inf X) where X is Subset of the complete Scott TopAugmentation of N : ( x in X & X in sigma the complete Scott TopAugmentation of N ) } , the complete Scott TopAugmentation of N) by
.= "\/" ( { (inf X) where X is Subset of the complete Scott TopAugmentation of N : ( x in X & X in sigma the complete Scott TopAugmentation of N ) } ,N) by
.= "\/" ( { (inf V) where V is Subset of N : ( x in V & V in lambda N ) } ,N) by Th34 ;
:: thesis: verum
end;
assume A4: for x being Element of N holds x = "\/" ( { (inf V) where V is Subset of N : ( x in V & V in lambda N ) } ,N) ; :: thesis: N is continuous
now :: thesis: for x being Element of the complete Scott TopAugmentation of N holds x = "\/" ( { (inf V) where V is Subset of the complete Scott TopAugmentation of N : ( x in V & V in sigma the complete Scott TopAugmentation of N ) } , the complete Scott TopAugmentation of N)
let x be Element of the complete Scott TopAugmentation of N; :: thesis: x = "\/" ( { (inf V) where V is Subset of the complete Scott TopAugmentation of N : ( x in V & V in sigma the complete Scott TopAugmentation of N ) } , the complete Scott TopAugmentation of N)
thus x = "\/" ( { (inf V) where V is Subset of N : ( x in V & V in lambda N ) } ,N) by A1, A4
.= "\/" ( { (inf V) where V is Subset of the complete Scott TopAugmentation of N : ( x in V & V in sigma the complete Scott TopAugmentation of N ) } ,N) by
.= "\/" ( { (inf V) where V is Subset of the complete Scott TopAugmentation of N : ( x in V & V in sigma the complete Scott TopAugmentation of N ) } , the complete Scott TopAugmentation of N) by ; :: thesis: verum
end;
then the complete Scott TopAugmentation of N is continuous by WAYBEL14:38;
hence N is continuous by ; :: thesis: verum