In reference to the next phase of the United States of America (Great Depression), other prospective concepts are exposed:� Continental China will invade the Unites States of America in the mentioned phase (possible and probable hypothesis). The Mathematics of History shows that in depressive phases, the great nuclei invade each other when they are in these phases within the general processes that follow, as well as other nations that may have a historical relationship due to borders, nearby zones or other interests, with the great nucleus in a depression. It is the dynamics of history, its development and process, the comings and goings of nations in their respective phases, is the oscillation of man, of cultures and civilizations, that reach their maximum and afterwards a minimum level. E. The Battle of Tours in the year 732, Charles Martel against the Arabs, the latter commanded by the brave Abderraman. It was a warlike action of the second aggressive process of the French Empire against the Arabs. The French Empire developed its first aggressive process from the year 250 to 561 (a duration of 311 years), its great depression from 561 to 688� (a duration of 125 years), the second period of great strategic victories from 688 to 815 (a duration of 129 years) and disintegration from 815 to 879 (64 years), resulting in a total of 629 years. The Arabs, owners of Spain and after one century of victories, attack Gaul in the year 732. After the defeat of Tours, they retreat to Spain for eight centuries until they return to their continent due to Hispanic weapons. F. The Battle of Saratoga (October 7, 1778). This battle decides the independence of the United States of America. Great Britain was practically at the beginning of its great depression when it faced its then colony in this decisive battle. G. The Battle of Lepanto (October 7, 1571). In the historical process of Europe and adjacent areas, where it concerns the first mathematical-historical cycle, in chronological order, the Persian, Carthaginian, Macedonian, Roman, Visigoth, French, and Teutonic Empires developed. It was the mathematical substitution of the supremacy of some for others respectively. The second mathematical-historical cycle begins with the Turkish Empire, then the Spanish, French, English Empires and the Federal Republic of Germany. The Turkish Empire is followed by the Spanish, so that the Battle of Lepanto, a warlike decisive event in this process, expresses the phase of decadence and disintegration of the Turkish Empire and the first period of great victories of the Spanish Empire. This is the reason for the result of this battle. �Great strategic victories belong to the phases corresponding to two great aggressive processes. Victory is a constant and it is the conclusion of every attack. That is the phase of great offensives�.� This was the phase of the Spanish Empire.