let T be TopSpace; :: thesis: for A, B being Element of Closed_Domains_of T holds (CLD-Union T) . A,B = (D-Union T) . A,B
let A, B be Element of Closed_Domains_of T; :: thesis: (CLD-Union T) . A,B = (D-Union T) . A,B
A1: ( A in { D where D is Subset of T : D is closed_condensed } & B in { E where E is Subset of T : E is closed_condensed } ) ;
then consider D being Subset of T such that
A2: ( D = A & D is closed_condensed ) ;
consider E being Subset of T such that
A3: ( E = B & E is closed_condensed ) by A1;
D \/ E is closed_condensed by A2, A3, TOPS_1:108;
then D \/ E is condensed by TOPS_1:106;
then A4: Int (Cl (A \/ B)) c= A \/ B by A2, A3, TOPS_1:def 6;
reconsider A0 = A, B0 = B as Element of Domains_of T ;
thus (CLD-Union T) . A,B = A \/ B by Def6
.= (Int (Cl (A0 \/ B0))) \/ (A0 \/ B0) by A4, XBOOLE_1:12
.= (D-Union T) . A,B by Def2 ; :: thesis: verum