let T be TopSpace; :: thesis: ( TopStruct(# the carrier of T, the topology of T #) is connected implies T is connected )
set G = TopStruct(# the carrier of T, the topology of T #);
assume A1: TopStruct(# the carrier of T, the topology of T #) is connected ; :: thesis: T is connected
let A, B be Subset of T; :: according to CONNSP_1:def 2 :: thesis: ( not [#] T = A \/ B or not A,B are_separated or A = {} T or B = {} T )
assume A2: ( [#] T = A \/ B & A,B are_separated ) ; :: thesis: ( A = {} T or B = {} T )
reconsider A1 = A, B1 = B as Subset of TopStruct(# the carrier of T, the topology of T #) ;
( [#] TopStruct(# the carrier of T, the topology of T #) = A1 \/ B1 & A1,B1 are_separated ) by A2, Th6;
then ( A1 = {} TopStruct(# the carrier of T, the topology of T #) or B1 = {} TopStruct(# the carrier of T, the topology of T #) ) by A1;
hence ( A = {} T or B = {} T ) ; :: thesis: verum