let n be Element of NAT ; :: thesis: for X, B1, B2 being Subset of (TOP-REAL n) st 0. (TOP-REAL n) in B1 holds
X (*) (B1,B2) c= X
let X, B1, B2 be Subset of (TOP-REAL n); :: thesis: ( 0. (TOP-REAL n) in B1 implies X (*) (B1,B2) c= X )
assume A1: 0. (TOP-REAL n) in B1 ; :: thesis: X (*) (B1,B2) c= X
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X (*) (B1,B2) or x in X )
assume 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
A2: x = y and
A3: 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 A2;
hence x in X by A3; :: thesis: verum