let T be non empty right_complementable Abelian add-associative right_zeroed RLSStruct ; :: thesis: for X, Y being Subset of T
for p being Point of T holds (X + p) (-) Y = (X (-) Y) + p
let X, Y be Subset of T; :: thesis: for p being Point of T holds (X + p) (-) Y = (X (-) Y) + p
let p be Point of T; :: 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 object ; :: 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 T 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 T : Y + y1 c= X } ;
then (y - p) + p in { (q + p) where q is Point of T : q in X (-) Y } ;
hence x in (X (-) Y) + p by A1, Lm2; :: thesis: verum
end;
let x be object ; :: 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 T such that
A3: x = y + p and
A4: y in X (-) Y ;
reconsider x = x as Point of T by A3;
( x - p = y & ex y2 being Point of T st
( y = y2 & Y + y2 c= X ) ) by A3, A4, Lm2;
then Y + x c= X + p by Th13;
hence x in (X + p) (-) Y ; :: thesis: verum