let E be RealLinearSpace; :: thesis: for A, B being Subset of E st B = {} holds
( A (+) B = B & B (+) A = B & A (-) B = the carrier of E )
let A, B be Subset of E; :: thesis: ( B = {} implies ( A (+) B = B & B (+) A = B & A (-) B = the carrier of E ) )
assume AS: B = {} ; :: thesis: ( A (+) B = B & B (+) A = B & A (-) B = the carrier of E )
hence ( A (+) B = B & B (+) A = B ) by RUSUB_5:5; :: thesis: A (-) B = the carrier of E

now :: thesis: for x being set st x in the carrier of E holds
x in A (-) B

let x be set ; :: thesis: ( x in the carrier of E implies x in A (-) B )
assume x in the carrier of E ; :: thesis: x in A (-) B
then reconsider z = x as Element of E ;
for b being Element of E st b in B holds
z - b in A by AS;
hence x in A (-) B ; :: thesis: verum

end;

then the carrier of E c= A (-) B by TARSKI:def 3;
hence the carrier of E = A (-) B by XBOOLE_0:def 10; :: thesis: verum