let X be non empty TopSpace; :: thesis: for X1, X2 being non empty V144() SubSpace of X st ( X1 is open or X2 is open ) holds

X1 union X2 is V144()

let X1, X2 be non empty V144() SubSpace of X; :: thesis: ( ( X1 is open or X2 is open ) implies X1 union X2 is V144() )

reconsider A1 = the carrier of X1, A2 = the carrier of X2 as non empty Subset of X by TSEP_1:1;

assume ( X1 is open or X2 is open ) ; :: thesis: X1 union X2 is V144()

then A1: ( A1 is open or A2 is open ) by TSEP_1:16;

( A1 is T_0 & A2 is T_0 ) by Th13;

then ( the carrier of (X1 union X2) = A1 \/ A2 & A1 \/ A2 is T_0 ) by A1, Th7, TSEP_1:def 2;

hence X1 union X2 is V144() ; :: thesis: verum

X1 union X2 is V144()

let X1, X2 be non empty V144() SubSpace of X; :: thesis: ( ( X1 is open or X2 is open ) implies X1 union X2 is V144() )

reconsider A1 = the carrier of X1, A2 = the carrier of X2 as non empty Subset of X by TSEP_1:1;

assume ( X1 is open or X2 is open ) ; :: thesis: X1 union X2 is V144()

then A1: ( A1 is open or A2 is open ) by TSEP_1:16;

( A1 is T_0 & A2 is T_0 ) by Th13;

then ( the carrier of (X1 union X2) = A1 \/ A2 & A1 \/ A2 is T_0 ) by A1, Th7, TSEP_1:def 2;

hence X1 union X2 is V144() ; :: thesis: verum