let n be Element of NAT ; :: thesis: for X, B, C being Subset of (TOP-REAL n) holds X (-) (B \/ C) = (X (-) B) /\ (X (-) C)
let X, B, C be Subset of (TOP-REAL n); :: thesis: X (-) (B \/ C) = (X (-) B) /\ (X (-) C)
thus X (-) (B \/ C) c= (X (-) B) /\ (X (-) C) :: according to XBOOLE_0:def 10 :: thesis: (X (-) B) /\ (X (-) C) c= X (-) (B \/ C)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X (-) (B \/ C) or x in (X (-) B) /\ (X (-) C) )
assume x in X (-) (B \/ C) ; :: thesis: x in (X (-) B) /\ (X (-) C)
then consider y being Point of (TOP-REAL n) such that
A1: ( x = y & (B \/ C) + y c= X ) ;
(B + y) \/ (C + y) c= X by A1, Th27;
then ( B + y c= X & C + y c= X ) by XBOOLE_1:11;
then ( x in { y1 where y1 is Point of (TOP-REAL n) : B + y1 c= X } & x in { y1 where y1 is Point of (TOP-REAL n) : C + y1 c= X } ) by A1;
hence x in (X (-) B) /\ (X (-) C) by XBOOLE_0:def 4; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (X (-) B) /\ (X (-) C) or x in X (-) (B \/ C) )
assume x in (X (-) B) /\ (X (-) C) ; :: thesis: x in X (-) (B \/ C)
then A2: ( x in X (-) B & x in X (-) C ) by XBOOLE_0:def 4;
then consider y being Point of (TOP-REAL n) such that
A3: ( x = y & B + y c= X ) ;
consider y2 being Point of (TOP-REAL n) such that
A4: ( x = y2 & C + y2 c= X ) by A2;
(B + y) \/ (C + y) c= X by A3, A4, XBOOLE_1:8;
then (B \/ C) + y c= X by Th27;
hence x in X (-) (B \/ C) by A3; :: thesis: verum