let E be RealLinearSpace; :: thesis: for A, B being Subset of E st B = the carrier of E holds
B (-) A = B
let A, B be Subset of E; :: thesis: ( B = the carrier of E implies B (-) A = B )
assume A1: B = the carrier of E ; :: thesis: B (-) A = B
now :: thesis: for x being object st x in the carrier of E holds
x in B (-) A
let x be object ; :: thesis: ( x in the carrier of E implies x in B (-) A )
assume x in the carrier of E ; :: thesis: x in B (-) A
then reconsider z = x as Element of E ;
for a being Element of E st a in A holds
z - a in B by A1;
hence x in B (-) A ; :: thesis: verum
end;
then the carrier of E c= B (-) A ;
hence B = B (-) A by A1; :: thesis: verum