let X be non empty TopSpace; :: thesis: for X1, X2 being non empty SubSpace of X st X1 meets X2 holds

for x being Point of (X1 meet X2) holds

( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x )

let X1, X2 be non empty SubSpace of X; :: thesis: ( X1 meets X2 implies for x being Point of (X1 meet X2) holds

( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x ) )

assume A1: X1 meets X2 ; :: thesis: for x being Point of (X1 meet X2) holds

( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x )

reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;

reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;

reconsider A0 = the carrier of (X1 meet X2) as Subset of X by TSEP_1:1;

let x be Point of (X1 meet X2); :: thesis: ( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x )

A0 = A1 /\ A2 by A1, TSEP_1:def 4;

then ( x in A1 & x in A2 ) by XBOOLE_0:def 4;

hence ( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x ) ; :: thesis: verum

for x being Point of (X1 meet X2) holds

( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x )

let X1, X2 be non empty SubSpace of X; :: thesis: ( X1 meets X2 implies for x being Point of (X1 meet X2) holds

( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x ) )

assume A1: X1 meets X2 ; :: thesis: for x being Point of (X1 meet X2) holds

( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x )

reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;

reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;

reconsider A0 = the carrier of (X1 meet X2) as Subset of X by TSEP_1:1;

let x be Point of (X1 meet X2); :: thesis: ( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x )

A0 = A1 /\ A2 by A1, TSEP_1:def 4;

then ( x in A1 & x in A2 ) by XBOOLE_0:def 4;

hence ( ex x1 being Point of X1 st x1 = x & ex x2 being Point of X2 st x2 = x ) ; :: thesis: verum