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

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