let X be non empty TopSpace; :: thesis: for X1, X2 being non empty T_0 SubSpace of X st ( X1 is closed or X2 is closed ) holds
X1 union X2 is T_0
let X1, X2 be non empty T_0 SubSpace of X; :: thesis: ( ( X1 is closed or X2 is closed ) implies X1 union X2 is T_0 )
reconsider A1 = the carrier of X1, A2 = the carrier of X2 as non empty Subset of X by TSEP_1:1;
assume ( X1 is closed or X2 is closed ) ; :: thesis: X1 union X2 is T_0
then A1: ( A1 is closed or A2 is closed ) by TSEP_1:11;
( 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, Th8, TSEP_1:def 2;
hence X1 union X2 is T_0 ; :: thesis: verum