let n be Element of NAT ; :: thesis: for X, B1, B2 being Subset of (TOP-REAL n) st 0. (TOP-REAL n) in B2 holds
(X (*) (B1,B2)) /\ X = {}
let X, B1, B2 be Subset of (TOP-REAL n); :: thesis: ( 0. (TOP-REAL n) in B2 implies (X (*) (B1,B2)) /\ X = {} )
assume A1: 0. (TOP-REAL n) in B2 ; :: thesis: (X (*) (B1,B2)) /\ X = {}

now :: thesis: for x being object holds not x in (X (*) (B1,B2)) /\ X

given x being object such that A2: x in (X (*) (B1,B2)) /\ X ; :: thesis: contradiction
A3: x in X by A2, XBOOLE_0:def 4;
x in X (*) (B1,B2) by A2, XBOOLE_0:def 4;
then x in (X `) (-) B2 by XBOOLE_0:def 4;
then consider y being Point of (TOP-REAL n) such that
A4: x = y and
A5: B2 + y c= X ` ;
(0. (TOP-REAL n)) + y in { (z + y) where z is Point of (TOP-REAL n) : z in B2 } by A1;
then x in B2 + y by A4;
hence contradiction by A3, A5, XBOOLE_0:def 5; :: thesis: verum

end;

hence (X (*) (B1,B2)) /\ X = {} by XBOOLE_0:def 1; :: thesis: verum