let N be complete Lawson TopLattice; :: thesis: for T being complete LATTICE
for A being correct Lawson TopAugmentation of T st RelStr(# the carrier of N, the InternalRel of N #) = RelStr(# the carrier of T, the InternalRel of T #) holds
TopRelStr(# the carrier of A, the InternalRel of A, the topology of A #) = TopRelStr(# the carrier of N, the InternalRel of N, the topology of N #)

let T be complete LATTICE; :: thesis: for A being correct Lawson TopAugmentation of T st RelStr(# the carrier of N, the InternalRel of N #) = RelStr(# the carrier of T, the InternalRel of T #) holds
TopRelStr(# the carrier of A, the InternalRel of A, the topology of A #) = TopRelStr(# the carrier of N, the InternalRel of N, the topology of N #)

let A be correct Lawson TopAugmentation of T; :: thesis: ( RelStr(# the carrier of N, the InternalRel of N #) = RelStr(# the carrier of T, the InternalRel of T #) implies TopRelStr(# the carrier of A, the InternalRel of A, the topology of A #) = TopRelStr(# the carrier of N, the InternalRel of N, the topology of N #) )
assume A1: RelStr(# the carrier of N, the InternalRel of N #) = RelStr(# the carrier of T, the InternalRel of T #) ; :: thesis: TopRelStr(# the carrier of A, the InternalRel of A, the topology of A #) = TopRelStr(# the carrier of N, the InternalRel of N, the topology of N #)
A2: omega T = omega N by ;
set S = the correct Scott TopAugmentation of T;
set l = the correct lower TopAugmentation of T;
A3: RelStr(# the carrier of the correct lower TopAugmentation of T, the InternalRel of the correct lower TopAugmentation of T #) = RelStr(# the carrier of T, the InternalRel of T #) by YELLOW_9:def 4;
A4: RelStr(# the carrier of the correct Scott TopAugmentation of T, the InternalRel of the correct Scott TopAugmentation of T #) = RelStr(# the carrier of T, the InternalRel of T #) by YELLOW_9:def 4;
the topology of the correct Scott TopAugmentation of T \/ the topology of the correct lower TopAugmentation of T c= bool the carrier of N
proof
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in the topology of the correct Scott TopAugmentation of T \/ the topology of the correct lower TopAugmentation of T or a in bool the carrier of N )
assume a in the topology of the correct Scott TopAugmentation of T \/ the topology of the correct lower TopAugmentation of T ; :: thesis: a in bool the carrier of N
then ( a in the topology of the correct Scott TopAugmentation of T or a in the topology of the correct lower TopAugmentation of T ) by XBOOLE_0:def 3;
hence a in bool the carrier of N by A1, A4, A3; :: thesis: verum
end;
then reconsider X = the topology of the correct Scott TopAugmentation of T \/ the topology of the correct lower TopAugmentation of T as Subset-Family of N ;
reconsider X = X as Subset-Family of N ;
A5: the topology of the correct lower TopAugmentation of T = omega T by WAYBEL19:def 2;
(sigma N) \/ () is prebasis of N by WAYBEL19:def 3;
then A6: (sigma T) \/ () is prebasis of N by ;
A7: the topology of the correct Scott TopAugmentation of T = sigma T by YELLOW_9:51;
the carrier of N = the carrier of the correct Scott TopAugmentation of T \/ the carrier of the correct lower TopAugmentation of T by A1, A4, A3;
then N is Refinement of the correct Scott TopAugmentation of T, the correct lower TopAugmentation of T by ;
then A8: the topology of N = UniCl () by YELLOW_9:56
.= lambda T by
.= the topology of A by WAYBEL19:def 4 ;
RelStr(# the carrier of A, the InternalRel of A #) = RelStr(# the carrier of N, the InternalRel of N #) by ;
hence TopRelStr(# the carrier of A, the InternalRel of A, the topology of A #) = TopRelStr(# the carrier of N, the InternalRel of N, the topology of N #) by A8; :: thesis: verum