381:
31:
629:
442:
536:
1695:
Rein analytischer Beweis des
Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewƤhren, wenigstens eine reelle Wurzel der Gleichung liege
468:
160:
74:
1276:
52:
1734:
Willard says that an ordered space "X is
Dedekind complete if every subset of X having an upper bound has a least upper bound." (pp. 124-5, Problem 17E.)
17:
707:
It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges. Let
688:
658:
takes advantage of this failure by defining the irrational numbers as the least upper bounds of certain subsets of the rationals.
390:
1939:
1916:
1874:
1855:
1725:
Bartle and
Sherbert (2011) define the "completeness property" and say that it is also called the "supremum property". (p. 39)
1889:
1958:
1897:
1834:
1812:
684:
651:
624:{\displaystyle \left\{x\in \mathbf {Q} :x^{2}\leq 2\right\}=\mathbf {Q} \cap \left(-{\sqrt {2}},{\sqrt {2}}\right)}
202:
174:
1195:
190:
1746:
1706:
475:
217:
1109:
368:
states that any non-empty set of real numbers that has an upper bound must have a least upper bound in
186:
1822:
1777:
680:
451:
143:
57:
1534:
692:
198:
699:, either directly from the construction or as a consequence of some other form of completeness.
1977:
1982:
1390:
485:
221:
194:
140:. Not every (partially) ordered set has the least upper bound property. For example, the set
105:
1255:
530:
does not have the least-upper-bound property under the usual order. For instance, the set
225:
8:
1361:
1083:
1769:
1744:
Raman-Sundstrƶm, Manya (AugustāSeptember 2015). "A Pedagogical
History of Compactness".
1781:
1773:
1755:
1119:
672:
37:
1954:
1935:
1928:
1912:
1893:
1870:
1851:
1830:
1808:
1785:
647:
128:
114:
1765:
1530:
723:
has exactly one element, then its only element is a least upper bound. So consider
380:
233:
205:, and it is also intimately related to the construction of the real numbers using
1934:. Walter Rudin Student Series in Advanced Mathematics (3 ed.). McGrawāHill.
1690:
676:
527:
163:
1971:
1061:
182:
655:
213:
206:
30:
1689:
The importance of the least-upper-bound property was first recognized by
1315:
1217:
480:
More generally, one may define upper bound and least upper bound for any
271:
256:
229:
124:
101:
81:
1548:
covers as well. This statement can be proved by considering the set
750:
is nonempty and has more than one element, there exists a real number
691:, the property is usually taken as an axiom for the real numbers (see
1122:
in the interval . This theorem can be proved by considering the set
1034:. It follows that both sequences are Cauchy and have the same limit
1526:
1205:
714:
671:
The least-upper-bound property is equivalent to other forms of the
316:
1760:
1807:. Undergraduate Texts in Mathematics. New York: Springer-Verlag.
696:
1906:
695:); in a constructive approach, the property must be proved as a
505:
has the least-upper-bound property if every non-empty subset of
1060:
can be used to prove many of the main foundational theorems in
481:
1216:
of real numbers in a closed interval must have a convergent
181:. It can be used to prove many of the fundamental results of
1162:
is the initial segment of that takes negative values under
437:{\displaystyle \left\{x\in \mathbf {Q} :x^{2}\leq 2\right\}}
1821:
177:
for the real numbers, and is sometimes referred to as
113:
has the least-upper-bound property if every non-empty
1258:
1220:. This theorem can be proved by considering the set
683:. The logical status of the property depends on the
539:
454:
393:
146:
60:
40:
1591:
by construction. By the least-upper-bound property,
76:
which is bounded from above has a least upper bound.
232:and has the least upper bound property is called a
1927:
1270:
623:
462:
436:
216:, this property can be generalized to a notion of
173:The least-upper-bound property is one form of the
154:
68:
46:
375:
1969:
511:with an upper bound has a least upper bound in
201:. It is usually taken as an axiom in synthetic
1883:
1864:
1846:Bartle, Robert G.; Sherbert, Donald R. (2011).
1802:
1743:
702:
1189:
1180:, and the least upper bound must be a root of
1925:
1850:(4 ed.). New York: John Wiley and Sons.
1420:. This can be proved by considering the set
1067:
729:with more than one element, and suppose that
493:, with āreal numberā replaced by āelement of
244:
239:
1907:Dangello, Frank; Seyfried, Michael (1999).
640:, but does not have a least upper bound in
1040:, which must be the least upper bound for
1759:
1537:states that some finite subcollection of
148:
62:
1737:
1560:ā ā:ā can be covered by finitely many
1345:
379:
29:
1948:
1486:, then it follows from continuity that
14:
1970:
1886:Mathematical Analysis: An Introduction
1629:that can be covered by finitely many
1610:is itself an element of some open set
1503:
1953:. Mineola, N.Y.: Dover Publications.
170:have the least upper bound property.
1336:has a subsequence that converges to
1930:Principles of Mathematical Analysis
1867:A Radical Approach to Real Analysis
1778:10.4169/amer.math.monthly.122.7.619
1770:10.4169/amer.math.monthly.122.7.619
893:Otherwise there must be an element
100:) is a fundamental property of the
27:Property of a partially ordered set
24:
1890:Undergraduate Texts in Mathematics
1054:The least-upper-bound property of
25:
1994:
666:
646:(since the square root of two is
203:constructions of the real numbers
685:construction of the real numbers
652:construction of the real numbers
584:
552:
456:
406:
1049:
759:that is not an upper bound for
448:the set of its upper bounds in
1728:
1719:
499:ā. In this case, we say that
376:Generalization to ordered sets
13:
1:
1892:. New York: Springer-Verlag.
1848:Introduction to Real Analysis
1796:
1747:American Mathematical Monthly
1508:Let be a closed interval in
1462:, and by its own definition,
675:, such as the convergence of
1829:(Third ed.). Academic.
1707:List of real analysis topics
1638:for some sufficiently small
1480:is the least upper bound of
703:Proof using Cauchy sequences
463:{\displaystyle \mathbf {Q} }
166:with its natural order does
155:{\displaystyle \mathbb {Q} }
69:{\displaystyle \mathbb {R} }
7:
1845:
1827:Principles of real analysis
1825:; Burkinshaw, Owen (1998).
1700:
1284:is not empty. In addition,
1196:BolzanoāWeierstrass theorem
1190:BolzanoāWeierstrass theorem
476:Completeness (order theory)
191:BolzanoāWeierstrass theorem
18:Least upper bound principle
10:
1999:
1949:Willard, Stephen (2004) .
1909:Introductory Real Analysis
1684:
1665:is not an upper bound for
1110:intermediate value theorem
1068:Intermediate value theorem
473:
366:least-upper-bound property
245:Statement for real numbers
187:intermediate value theorem
86:least-upper-bound property
1823:Aliprantis, Charalambos D
1389:has no upper bound. The
240:Statement of the property
1884:Browder, Andrew (1996).
1865:Bressoud, David (2007).
1803:Abbott, Stephen (2001).
1712:
1597:has a least upper bound
1302:has a least upper bound
813:recursively as follows:
717:set of real numbers. If
681:nested intervals theorem
661:
693:least upper bound axiom
34:Every non-empty subset
1926:Rudin, Walter (1976).
1805:Understanding Analysis
1327:, and it follows that
1290:is an upper bound for
1272:
1271:{\displaystyle a\in S}
1174:is an upper bound for
835:is an upper bound for
634:has an upper bound in
625:
471:
464:
438:
348:for every upper bound
332:is an upper bound for
255:be a non-empty set of
156:
77:
70:
48:
1619:, and it follows for
1391:extreme value theorem
1346:Extreme value theorem
1273:
1108:. In this case, the
626:
520:For example, the set
486:partially ordered set
465:
439:
383:
222:partially ordered set
195:extreme value theorem
179:Dedekind completeness
157:
106:partially ordered set
71:
49:
33:
1585:, and is bounded by
1256:
1243:for infinitely many
765:. Define sequences
537:
452:
391:
226:linearly ordered set
144:
104:. More generally, a
58:
54:of the real numbers
38:
1645:. This proves that
1579:obviously contains
1535:HeineāBorel theorem
1525:be a collection of
1504:HeineāBorel theorem
1362:continuous function
1086:, and suppose that
1084:continuous function
735:has an upper bound
199:HeineāBorel theorem
1693:in his 1817 paper
1268:
1204:states that every
689:synthetic approach
673:completeness axiom
621:
472:
460:
434:
175:completeness axiom
152:
88:(sometimes called
78:
66:
44:
1941:978-0-07-054235-8
1918:978-0-395-95933-6
1876:978-0-88385-747-2
1857:978-0-471-43331-6
1446:By definition of
1142:) < 0 for all
614:
604:
312:least upper bound
94:supremum property
47:{\displaystyle M}
16:(Redirected from
1990:
1964:
1951:General Topology
1945:
1933:
1922:
1903:
1880:
1861:
1840:
1818:
1790:
1789:
1763:
1741:
1735:
1732:
1726:
1723:
1680:
1671:. Consequently,
1670:
1664:
1658:
1644:
1637:
1628:
1618:
1609:
1603:
1596:
1590:
1584:
1578:
1568:
1547:
1524:
1513:
1499:
1485:
1479:
1473:
1467:
1461:
1451:
1441:
1419:
1412:
1398:
1388:
1381:
1374:
1359:
1341:
1335:
1326:
1318:of the sequence
1313:
1307:
1301:
1295:
1289:
1283:
1277:
1275:
1274:
1269:
1248:
1215:
1203:
1185:
1179:
1173:
1167:
1161:
1151:
1117:
1107:
1096:
1081:
1059:
1045:
1039:
1033:
1026:
1008:
960:
941:
925:
904:
898:
889:
862:
840:
834:
812:
788:
764:
758:
749:
743:
734:
728:
722:
712:
677:Cauchy sequences
645:
639:
630:
628:
627:
622:
620:
616:
615:
610:
605:
600:
587:
579:
575:
568:
567:
555:
528:rational numbers
525:
516:
510:
504:
498:
492:
469:
467:
466:
461:
459:
443:
441:
440:
435:
433:
429:
422:
421:
409:
359:
353:
347:
337:
331:
325:
309:
300:
290:
280:
268:
254:
234:linear continuum
164:rational numbers
161:
159:
158:
153:
151:
139:
122:
112:
75:
73:
72:
67:
65:
53:
51:
50:
45:
21:
1998:
1997:
1993:
1992:
1991:
1989:
1988:
1987:
1968:
1967:
1961:
1942:
1919:
1911:. Brooks Cole.
1900:
1877:
1858:
1837:
1815:
1799:
1794:
1793:
1742:
1738:
1733:
1729:
1724:
1720:
1715:
1703:
1691:Bernard Bolzano
1687:
1672:
1666:
1660:
1646:
1639:
1635:
1630:
1620:
1616:
1611:
1605:
1598:
1592:
1586:
1580:
1574:
1565:
1552:
1544:
1538:
1521:
1515:
1509:
1506:
1487:
1481:
1475:
1469:
1468:is bounded by
1463:
1453:
1447:
1424:
1414:
1400:
1394:
1383:
1376:
1365:
1351:
1348:
1337:
1333:
1328:
1324:
1319:
1309:
1303:
1297:
1291:
1285:
1279:
1257:
1254:
1253:
1241:
1224:
1213:
1208:
1199:
1192:
1181:
1175:
1169:
1163:
1157:
1126:
1113:
1098:
1087:
1073:
1070:
1055:
1052:
1041:
1035:
1028:
1023:
1016:
1010:
1007:
1000:
993:
986:
979:
972:
966:
958:
952:
943:
936:
927:
922:
915:
906:
900:
894:
886:
879:
873:
864:
860:
854:
845:
836:
831:
824:
818:
810:
803:
796:
790:
786:
779:
772:
766:
760:
757:
751:
745:
742:
736:
730:
724:
718:
708:
705:
669:
664:
641:
635:
609:
599:
595:
591:
583:
563:
559:
551:
544:
540:
538:
535:
534:
521:
512:
506:
500:
494:
488:
478:
455:
453:
450:
449:
417:
413:
405:
398:
394:
392:
389:
388:
378:
355:
349:
339:
333:
327:
321:
305:
292:
282:
276:
264:
250:
247:
242:
147:
145:
142:
141:
135:
118:
108:
98:l.u.b. property
61:
59:
56:
55:
39:
36:
35:
28:
23:
22:
15:
12:
11:
5:
1996:
1986:
1985:
1980:
1966:
1965:
1959:
1946:
1940:
1923:
1917:
1904:
1898:
1881:
1875:
1862:
1856:
1842:
1841:
1835:
1819:
1813:
1798:
1795:
1792:
1791:
1754:(7): 619ā635.
1736:
1727:
1717:
1716:
1714:
1711:
1710:
1709:
1702:
1699:
1686:
1683:
1633:
1614:
1571:
1570:
1563:
1542:
1519:
1505:
1502:
1444:
1443:
1399:is finite and
1347:
1344:
1331:
1322:
1267:
1264:
1261:
1250:
1249:
1239:
1211:
1191:
1188:
1154:
1153:
1069:
1066:
1051:
1048:
1021:
1014:
1005:
998:
991:
984:
977:
970:
963:
962:
956:
947:
931:
920:
913:
891:
884:
877:
868:
858:
849:
844:If it is, let
842:
829:
822:
817:Check whether
808:
801:
794:
784:
777:
770:
755:
740:
704:
701:
668:
667:Logical status
665:
663:
660:
632:
631:
619:
613:
608:
603:
598:
594:
590:
586:
582:
578:
574:
571:
566:
562:
558:
554:
550:
547:
543:
474:Main article:
458:
432:
428:
425:
420:
416:
412:
408:
404:
401:
397:
377:
374:
362:
361:
304:A real number
302:
263:A real number
246:
243:
241:
238:
185:, such as the
150:
134:(supremum) in
64:
43:
26:
9:
6:
4:
3:
2:
1995:
1984:
1981:
1979:
1978:Real analysis
1976:
1975:
1973:
1962:
1960:9780486434797
1956:
1952:
1947:
1943:
1937:
1932:
1931:
1924:
1920:
1914:
1910:
1905:
1901:
1899:0-387-94614-4
1895:
1891:
1887:
1882:
1878:
1872:
1868:
1863:
1859:
1853:
1849:
1844:
1843:
1838:
1836:0-12-050257-7
1832:
1828:
1824:
1820:
1816:
1814:0-387-95060-5
1810:
1806:
1801:
1800:
1787:
1783:
1779:
1775:
1771:
1767:
1762:
1757:
1753:
1749:
1748:
1740:
1731:
1722:
1718:
1708:
1705:
1704:
1698:
1696:
1692:
1682:
1679:
1675:
1669:
1663:
1657:
1653:
1649:
1642:
1636:
1627:
1623:
1617:
1608:
1601:
1595:
1589:
1583:
1577:
1566:
1559:
1555:
1551:
1550:
1549:
1545:
1536:
1532:
1528:
1522:
1512:
1501:
1498:
1494:
1490:
1484:
1478:
1472:
1466:
1460:
1456:
1450:
1439:
1435:
1431:
1427:
1423:
1422:
1421:
1417:
1411:
1407:
1403:
1397:
1392:
1386:
1379:
1372:
1368:
1363:
1358:
1354:
1343:
1340:
1334:
1325:
1317:
1312:
1306:
1300:
1294:
1288:
1282:
1265:
1262:
1259:
1246:
1242:
1235:
1231:
1227:
1223:
1222:
1221:
1219:
1214:
1207:
1202:
1197:
1187:
1184:
1178:
1172:
1166:
1160:
1149:
1145:
1141:
1137:
1133:
1129:
1125:
1124:
1123:
1121:
1116:
1111:
1105:
1101:
1094:
1090:
1085:
1080:
1076:
1065:
1063:
1062:real analysis
1058:
1047:
1044:
1038:
1031:
1024:
1017:
1004:
997:
990:
983:
976:
969:
959:
950:
946:
940:
934:
930:
923:
916:
909:
903:
897:
892:
887:
880:
871:
867:
861:
852:
848:
843:
839:
832:
825:
816:
815:
814:
807:
800:
793:
783:
776:
769:
763:
754:
748:
739:
733:
727:
721:
716:
711:
700:
698:
694:
690:
687:used: in the
686:
682:
678:
674:
659:
657:
656:Dedekind cuts
653:
649:
644:
638:
617:
611:
606:
601:
596:
592:
588:
580:
576:
572:
569:
564:
560:
556:
548:
545:
541:
533:
532:
531:
529:
524:
518:
515:
509:
503:
497:
491:
487:
483:
477:
447:
430:
426:
423:
418:
414:
410:
402:
399:
395:
386:
382:
373:
371:
367:
358:
352:
346:
342:
336:
330:
324:
319:
318:
313:
308:
303:
299:
295:
289:
285:
279:
274:
273:
269:is called an
267:
262:
261:
260:
258:
253:
237:
235:
231:
227:
223:
219:
215:
210:
208:
207:Dedekind cuts
204:
200:
196:
192:
188:
184:
183:real analysis
180:
176:
171:
169:
165:
138:
133:
131:
126:
121:
116:
111:
107:
103:
99:
95:
91:
87:
83:
41:
32:
19:
1983:Order theory
1950:
1929:
1908:
1885:
1866:
1847:
1826:
1804:
1751:
1745:
1739:
1730:
1721:
1694:
1688:
1677:
1673:
1667:
1661:
1655:
1651:
1647:
1640:
1631:
1625:
1621:
1612:
1606:
1599:
1593:
1587:
1581:
1575:
1572:
1561:
1557:
1553:
1540:
1533:. Then the
1517:
1510:
1507:
1496:
1492:
1488:
1482:
1476:
1470:
1464:
1458:
1454:
1448:
1445:
1437:
1433:
1429:
1425:
1415:
1409:
1405:
1401:
1395:
1393:states that
1384:
1377:
1370:
1366:
1356:
1352:
1349:
1338:
1329:
1320:
1310:
1304:
1298:
1292:
1286:
1280:
1251:
1244:
1237:
1233:
1229:
1225:
1209:
1200:
1193:
1182:
1176:
1170:
1164:
1158:
1155:
1147:
1143:
1139:
1135:
1131:
1127:
1118:must have a
1114:
1112:states that
1103:
1099:
1092:
1088:
1078:
1074:
1071:
1056:
1053:
1050:Applications
1042:
1036:
1029:
1019:
1012:
1002:
995:
988:
981:
974:
967:
964:
954:
948:
944:
938:
932:
928:
918:
911:
907:
901:
895:
882:
875:
869:
865:
856:
850:
846:
837:
827:
820:
805:
798:
791:
781:
774:
767:
761:
752:
746:
737:
731:
725:
719:
709:
706:
670:
642:
636:
633:
522:
519:
513:
507:
501:
495:
489:
479:
445:
384:
370:real numbers
369:
365:
363:
356:
350:
344:
340:
334:
328:
322:
315:
311:
306:
297:
293:
287:
283:
277:
270:
265:
257:real numbers
251:
248:
218:completeness
214:order theory
211:
178:
172:
167:
136:
129:
119:
109:
102:real numbers
97:
93:
90:completeness
89:
85:
79:
1604:. Hence,
1432:ā ā:ā sup
1316:limit point
1218:subsequence
272:upper bound
132:upper bound
125:upper bound
82:mathematics
1972:Categories
1797:References
1514:, and let
1314:must be a
648:irrational
197:, and the
1786:119936587
1761:1006.4131
1527:open sets
1413:for some
1263:∈
1252:Clearly,
1156:That is,
597:−
589:∩
570:≤
549:∈
424:≤
403:∈
1701:See also
1573:The set
1375:, where
1364:and let
1206:sequence
1168:. Then
1106:) > 0
1095:) < 0
942:and let
905:so that
863:and let
744:. Since
715:nonempty
650:). The
387:the set
317:supremum
291:for all
228:that is
220:for any
123:with an
1869:. MAA.
1685:History
1308:. Then
1232:ā ā:
1134:ā ā:
926:. Let
697:theorem
679:or the
310:is the
162:of all
1957:
1938:
1915:
1896:
1873:
1854:
1833:
1811:
1784:
1776:
1643:> 0
1531:covers
1369:= sup
1278:, and
987:ā¤ āÆ ā¤
654:using
482:subset
320:) for
224:. A
193:, the
189:, the
127:has a
115:subset
84:, the
1782:S2CID
1774:JSTOR
1756:arXiv
1713:Notes
1624:<
1529:that
1474:. If
1436:() =
1360:be a
1355:: ā
1296:, so
1082:be a
1077:: ā
1025:| ā 0
965:Then
924:) ā 2
910:>(
888:) ā 2
833:) ā 2
811:, ...
787:, ...
713:be a
662:Proof
484:of a
446:Blue:
230:dense
130:least
1955:ISBN
1936:ISBN
1913:ISBN
1894:ISBN
1871:ISBN
1852:ISBN
1831:ISBN
1809:ISBN
1659:and
1556:=ā {
1495:) =
1428:=ā {
1408:) =
1350:Let
1228:=ā {
1198:for
1194:The
1130:=ā {
1120:root
1097:and
1072:Let
1009:and
789:and
385:Red:
364:The
338:and
314:(or
275:for
249:Let
1766:doi
1752:122
1382:if
1380:= ā
1032:ā ā
1027:as
899:in
874:= (
526:of
354:of
326:if
281:if
212:In
209:.
168:not
117:of
96:or
80:In
1974::
1888:.
1780:.
1772:.
1764:.
1750:.
1697:.
1681:.
1676:=
1654:ā
1650:+
1602:ā
1567:}
1546:}
1523:}
1500:.
1457:ā
1452:,
1440:}
1418:ā
1387:()
1373:()
1342:.
1247:}
1236:ā¤
1186:.
1150:}
1146:ā¤
1064:.
1046:.
1018:ā
1001:ā¤
994:ā¤
980:ā¤
973:ā¤
953:=
951:+1
937:=
935:+1
917:+
881:+
872:+1
855:=
853:+1
826:+
804:,
797:,
780:,
773:,
517:.
444:.
372:.
343:ā¤
296:ā
286:ā„
259:.
236:.
92:,
1963:.
1944:.
1921:.
1902:.
1879:.
1860:.
1839:.
1817:.
1788:.
1768::
1758::
1678:b
1674:c
1668:S
1662:c
1656:S
1652:Ī“
1648:c
1641:Ī“
1634:Ī±
1632:U
1626:b
1622:c
1615:Ī±
1613:U
1607:c
1600:c
1594:S
1588:b
1582:a
1576:S
1569:.
1564:Ī±
1562:U
1558:s
1554:S
1543:Ī±
1541:U
1539:{
1520:Ī±
1518:U
1516:{
1511:R
1497:M
1493:c
1491:(
1489:f
1483:S
1477:c
1471:b
1465:S
1459:S
1455:a
1449:M
1442:.
1438:M
1434:f
1430:s
1426:S
1416:c
1410:M
1406:c
1404:(
1402:f
1396:M
1385:f
1378:M
1371:f
1367:M
1357:R
1353:f
1339:c
1332:n
1330:x
1323:n
1321:x
1311:c
1305:c
1299:S
1293:S
1287:b
1281:S
1266:S
1260:a
1245:n
1240:n
1238:x
1234:s
1230:s
1226:S
1212:n
1210:x
1201:R
1183:f
1177:S
1171:b
1165:f
1159:S
1152:.
1148:s
1144:x
1140:x
1138:(
1136:f
1132:s
1128:S
1115:f
1104:b
1102:(
1100:f
1093:a
1091:(
1089:f
1079:R
1075:f
1057:R
1043:S
1037:L
1030:n
1022:n
1020:B
1015:n
1013:A
1011:|
1006:1
1003:B
999:2
996:B
992:3
989:B
985:3
982:A
978:2
975:A
971:1
968:A
961:.
957:n
955:B
949:n
945:B
939:s
933:n
929:A
921:n
919:B
914:n
912:A
908:s
902:S
896:s
890:.
885:n
883:B
878:n
876:A
870:n
866:B
859:n
857:A
851:n
847:A
841:.
838:S
830:n
828:B
823:n
821:A
819:(
809:3
806:B
802:2
799:B
795:1
792:B
785:3
782:A
778:2
775:A
771:1
768:A
762:S
756:1
753:A
747:S
741:1
738:B
732:S
726:S
720:S
710:S
643:Q
637:Q
618:)
612:2
607:,
602:2
593:(
585:Q
581:=
577:}
573:2
565:2
561:x
557::
553:Q
546:x
542:{
523:Q
514:X
508:X
502:X
496:X
490:X
470:.
457:Q
431:}
427:2
419:2
415:x
411::
407:Q
400:x
396:{
360:.
357:S
351:y
345:y
341:x
335:S
329:x
323:S
307:x
301:.
298:S
294:s
288:s
284:x
278:S
266:x
252:S
149:Q
137:X
120:X
110:X
63:R
42:M
20:)
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