let n be Element of NAT ; :: thesis: for X, Y being Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n) holds X (-) (Y + p) = (X (-) Y) + (- p)
let X, Y be Subset of (TOP-REAL n); :: thesis: for p being Point of (TOP-REAL n) holds X (-) (Y + p) = (X (-) Y) + (- p)
let p be Point of (TOP-REAL n); :: thesis: X (-) (Y + p) = (X (-) Y) + (- p)
thus X (-) (Y + p) c= (X (-) Y) + (- p) :: according to XBOOLE_0:def 10 :: thesis: (X (-) Y) + (- p) c= X (-) (Y + p)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X (-) (Y + p) or x in (X (-) Y) + (- p) )
assume x in X (-) (Y + p) ; :: thesis: x in (X (-) Y) + (- p)
then consider y being Point of (TOP-REAL n) such that
A1: x = y and
A2: (Y + p) + y c= X ;
Y + (y + p) c= X by A2, Th16;
then y + p in { y1 where y1 is Point of (TOP-REAL n) : Y + y1 c= X } ;
then (y + p) - p in (X (-) Y) + (- p) ;
hence x in (X (-) Y) + (- p) by A1, EUCLID:52; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (X (-) Y) + (- p) or x in X (-) (Y + p) )
assume x in (X (-) Y) + (- p) ; :: thesis: x in X (-) (Y + p)
then consider y being Point of (TOP-REAL n) such that
A3: x = y + (- p) and
A4: y in X (-) Y ;
reconsider x = x as Point of (TOP-REAL n) by A3;
x + p = (y - p) + p by A3;
then A5: x + p = y by EUCLID:52;
ex y2 being Point of (TOP-REAL n) st
( y = y2 & Y + y2 c= X ) by A4;
then (Y + p) + x c= X by A5, Th16;
hence x in X (-) (Y + p) ; :: thesis: verum