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 A1: 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 object st x in the carrier of E holds
x in A (-) B
let x be object ; :: 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 A1;
hence x in A (-) B ; :: thesis: verum
end;
then the carrier of E c= A (-) B ;
hence the carrier of E = A (-) B ; :: thesis: verum