let n be Element of NAT ; :: thesis: for B1, X, B2 being Subset of (TOP-REAL n) st 0. (TOP-REAL n) in B1 holds
X (&) B1,B2 = X
let B1, X, B2 be Subset of (TOP-REAL n); :: thesis: ( 0. (TOP-REAL n) in B1 implies X (&) B1,B2 = X )
assume A1: 0. (TOP-REAL n) in B1 ; :: thesis: X (&) B1,B2 = X
thus X (&) B1,B2 c= X :: according to XBOOLE_0:def 10 :: thesis: X c= X (&) B1,B2

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X (&) B1,B2 or x in X )
assume A2: x in X (&) B1,B2 ; :: thesis: x in X

per cases ( x in X or x in X (*) B1,B2 ) by A2, XBOOLE_0:def 3;

suppose x in X ; :: thesis: x in X

hence x in X ; :: thesis: verum

end;

suppose x in X (*) B1,B2 ; :: thesis: x in X

then x in X (-) B1 by XBOOLE_0:def 4;
then consider y being Point of (TOP-REAL n) such that
A3: x = y and
A4: B1 + y c= X ;
(0. (TOP-REAL n)) + y in { (z + y) where z is Point of (TOP-REAL n) : z in B1 } by A1;
then x in B1 + y by A3, EUCLID:31;
hence x in X by A4; :: thesis: verum

end;

end;

end;

thus X c= X (&) B1,B2 by XBOOLE_1:7; :: thesis: verum