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

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (X + p) (-) Y or x in (X (-) Y) + p )
assume x in (X + p) (-) Y ; :: thesis: x in (X (-) Y) + p
then consider y being Point of such that
A1: x = y and
A2: Y + y c= X + p ;
Y + (y - p) c= X by A2, Th13;
then y - p in { y1 where y1 is Point of : Y + y1 c= X } ;
then (y - p) + p in { (q + p) where q is Point of : q in X (-) Y } ;
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 + p) (-) Y )
assume x in (X (-) Y) + p ; :: thesis: x in (X + p) (-) Y
then consider y being Point of such that
A3: x = y + p and
A4: y in X (-) Y ;
reconsider x = x as Point of by A3;
( x - p = y & ex y2 being Point of st
( y = y2 & Y + y2 c= X ) ) by A3, A4, EUCLID:52;
then Y + x c= X + p by Th13;
hence x in (X + p) (-) Y ; :: thesis: verum