let it1, it2 be ManySortedSet of X; :: thesis: ( ( for x being object holds it1 . x = (b1 . x) -' (b2 . x) ) & ( for x being object holds it2 . x = (b1 . x) -' (b2 . x) ) implies it1 = it2 )
assume that
A5: for x being object holds it1 . x = (b1 . x) -' (b2 . x) and
A6: for x being object holds it2 . x = (b1 . x) -' (b2 . x) ; :: thesis: it1 = it2
now :: thesis: for x being object st x in X holds
it1 . x = it2 . x
let x be object ; :: thesis: ( x in X implies it1 . x = it2 . x )
assume x in X ; :: thesis: it1 . x = it2 . x
thus it1 . x = (b1 . x) -' (b2 . x) by A5
.= it2 . x by A6 ; :: thesis: verum
end;
hence it1 = it2 ; :: thesis: verum