let X be non empty TopSpace; :: thesis: for X1 being non empty V144() SubSpace of X

for X2 being non empty SubSpace of X st X1 meets X2 holds

X1 meet X2 is V144()

let X1 be non empty V144() SubSpace of X; :: thesis: for X2 being non empty SubSpace of X st X1 meets X2 holds

X1 meet X2 is V144()

let X2 be non empty SubSpace of X; :: thesis: ( X1 meets X2 implies X1 meet X2 is V144() )

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 V144()

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 V144() by A1; :: thesis: verum

for X2 being non empty SubSpace of X st X1 meets X2 holds

X1 meet X2 is V144()

let X1 be non empty V144() SubSpace of X; :: thesis: for X2 being non empty SubSpace of X st X1 meets X2 holds

X1 meet X2 is V144()

let X2 be non empty SubSpace of X; :: thesis: ( X1 meets X2 implies X1 meet X2 is V144() )

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 V144()

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 V144() by A1; :: thesis: verum