let it1, it2 be bag of the carrier of L; :: thesis: ( support it1 = Roots p & ( for x being Element of L holds it1 . x = multiplicity (p,x) ) & support it2 = Roots p & ( for x being Element of L holds it2 . x = multiplicity (p,x) ) implies it1 = it2 )
assume that
support it1 = Roots p and
A7: for x being Element of L holds it1 . x = multiplicity (p,x) and
support it2 = Roots p and
A8: for x being Element of L holds it2 . x = multiplicity (p,x) ; :: thesis: it1 = it2
now :: thesis: for x being object st x in the carrier of L holds
it1 . x = it2 . x
let x be object ; :: thesis: ( x in the carrier of L implies it1 . x = it2 . x )
assume x in the carrier of L ; :: thesis: it1 . x = it2 . x
then reconsider xx = x as Element of L ;
thus it1 . x = multiplicity (p,xx) by A7
.= it2 . x by A8 ; :: thesis: verum
end;
hence it1 = it2 by PBOOLE:3; :: thesis: verum