let X be non empty TopSpace; :: thesis: for X1 being non empty T_0 SubSpace of X
for X2 being non empty SubSpace of X st X1 meets X2 holds
X1 meet X2 is T_0
let X1 be non empty T_0 SubSpace of X; :: thesis: for X2 being non empty SubSpace of X st X1 meets X2 holds
X1 meet X2 is T_0
let X2 be non empty SubSpace of X; :: thesis: ( X1 meets X2 implies X1 meet X2 is T_0 )
reconsider A1 = the carrier of X1 as non empty Subset of X by TSEP_1:1;
reconsider A2 = the carrier of X2 as non empty Subset of X by TSEP_1:1;
assume X1 meets X2 ; :: thesis: X1 meet X2 is T_0
then A1: the carrier of (X1 meet X2) = A1 /\ A2 by TSEP_1:def 4;
A1 is T_0 by Th13;
then A1 /\ A2 is T_0 by Th6;
hence X1 meet X2 is T_0 by A1; :: thesis: verum