let n be Element of NAT ; :: thesis: for X being Subset of (TOP-REAL n) holds X (-) {(0. (TOP-REAL n))} = X
let X be Subset of (TOP-REAL n); :: thesis: X (-) {(0. (TOP-REAL n))} = X
thus X (-) {(0. (TOP-REAL n))} c= X :: according to XBOOLE_0:def 10 :: thesis: X c= X (-) {(0. (TOP-REAL n))}

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X (-) {(0. (TOP-REAL n))} or x in X )
assume x in X (-) {(0. (TOP-REAL n))} ; :: thesis: x in X
then ex y being Point of (TOP-REAL n) st
( x = y & {(0. (TOP-REAL n))} + y c= X ) ;
then {x} c= X by Th2;
hence x in X by ZFMISC_1:37; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in X (-) {(0. (TOP-REAL n))} )
assume A1: x in X ; :: thesis: x in X (-) {(0. (TOP-REAL n))}
then reconsider xx = x as Point of (TOP-REAL n) ;
{x} c= X by A1, ZFMISC_1:37;
then {(0. (TOP-REAL n))} + xx c= X by Th2;
hence x in X (-) {(0. (TOP-REAL n))} ; :: thesis: verum