let Y0, Y1 be TopStruct ; :: thesis: for D0 being Subset of Y0

for D1 being Subset of Y1 st TopStruct(# the carrier of Y0, the topology of Y0 #) = TopStruct(# the carrier of Y1, the topology of Y1 #) & D0 = D1 & D0 is maximal_anti-discrete holds

D1 is maximal_anti-discrete

let D0 be Subset of Y0; :: thesis: for D1 being Subset of Y1 st TopStruct(# the carrier of Y0, the topology of Y0 #) = TopStruct(# the carrier of Y1, the topology of Y1 #) & D0 = D1 & D0 is maximal_anti-discrete holds

D1 is maximal_anti-discrete

let D1 be Subset of Y1; :: thesis: ( TopStruct(# the carrier of Y0, the topology of Y0 #) = TopStruct(# the carrier of Y1, the topology of Y1 #) & D0 = D1 & D0 is maximal_anti-discrete implies D1 is maximal_anti-discrete )

assume A1: TopStruct(# the carrier of Y0, the topology of Y0 #) = TopStruct(# the carrier of Y1, the topology of Y1 #) ; :: thesis: ( not D0 = D1 or not D0 is maximal_anti-discrete or D1 is maximal_anti-discrete )

assume A2: D0 = D1 ; :: thesis: ( not D0 is maximal_anti-discrete or D1 is maximal_anti-discrete )

assume A3: D0 is maximal_anti-discrete ; :: thesis: D1 is maximal_anti-discrete

then D1 is anti-discrete by A1, A2, Th4;

hence D1 is maximal_anti-discrete by A4; :: thesis: verum

for D1 being Subset of Y1 st TopStruct(# the carrier of Y0, the topology of Y0 #) = TopStruct(# the carrier of Y1, the topology of Y1 #) & D0 = D1 & D0 is maximal_anti-discrete holds

D1 is maximal_anti-discrete

let D0 be Subset of Y0; :: thesis: for D1 being Subset of Y1 st TopStruct(# the carrier of Y0, the topology of Y0 #) = TopStruct(# the carrier of Y1, the topology of Y1 #) & D0 = D1 & D0 is maximal_anti-discrete holds

D1 is maximal_anti-discrete

let D1 be Subset of Y1; :: thesis: ( TopStruct(# the carrier of Y0, the topology of Y0 #) = TopStruct(# the carrier of Y1, the topology of Y1 #) & D0 = D1 & D0 is maximal_anti-discrete implies D1 is maximal_anti-discrete )

assume A1: TopStruct(# the carrier of Y0, the topology of Y0 #) = TopStruct(# the carrier of Y1, the topology of Y1 #) ; :: thesis: ( not D0 = D1 or not D0 is maximal_anti-discrete or D1 is maximal_anti-discrete )

assume A2: D0 = D1 ; :: thesis: ( not D0 is maximal_anti-discrete or D1 is maximal_anti-discrete )

assume A3: D0 is maximal_anti-discrete ; :: thesis: D1 is maximal_anti-discrete

A4: now :: thesis: for D being Subset of Y1 st D is anti-discrete & D1 c= D holds

D1 = D

D0 is anti-discrete
by A3;D1 = D

let D be Subset of Y1; :: thesis: ( D is anti-discrete & D1 c= D implies D1 = D )

reconsider E = D as Subset of Y0 by A1;

assume D is anti-discrete ; :: thesis: ( D1 c= D implies D1 = D )

then A5: E is anti-discrete by A1, Th4;

assume D1 c= D ; :: thesis: D1 = D

hence D1 = D by A2, A3, A5; :: thesis: verum

end;reconsider E = D as Subset of Y0 by A1;

assume D is anti-discrete ; :: thesis: ( D1 c= D implies D1 = D )

then A5: E is anti-discrete by A1, Th4;

assume D1 c= D ; :: thesis: D1 = D

hence D1 = D by A2, A3, A5; :: thesis: verum

then D1 is anti-discrete by A1, A2, Th4;

hence D1 is maximal_anti-discrete by A4; :: thesis: verum