def METHOD_NAME(afi, afid, bfi, bfid):
    """Merge two pairs of (index ids, index dimensions) sequences into one pair without duplicates.
    The id tuples afi, bfi are assumed already sorted by id.
    Given a list of (id, dim) tuples already sorted by id,
    return a unique list with duplicates removed.
    Also checks that the dimensions of duplicates are matching.
    """
    na = len(afi)
    nb = len(bfi)
    if na == 0:
        return bfi, bfid
    elif nb == 0:
        return afi, afid
    ak = 0
    bk = 0
    fi = []
    fid = []
    while True:
        if afi[ak] < bfi[bk]:
            fi.append(afi[ak])
            fid.append(afid[ak])
            ak += 1
        elif afi[ak] > bfi[bk]:
            fi.append(bfi[bk])
            fid.append(bfid[bk])
            bk += 1
        else:
            fi.append(afi[ak])
            fid.append(afid[ak])
            ak += 1
            bk += 1
        if ak == na:
            if bk != nb:
                fi.extend(bfi[bk:])
                fid.extend(bfid[bk:])
            break
        elif bk == nb:
            fi.extend(afi[ak:])
            fid.extend(afid[ak:])
            break
    return tuple(fi), tuple(fid)