now :: thesis: ( not 2 divides 7583 & not 3 divides 7583 & not 5 divides 7583 & not 7 divides 7583 & not 11 divides 7583 & not 13 divides 7583 & not 17 divides 7583 & not 19 divides 7583 & not 23 divides 7583 & not 29 divides 7583 & not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )

7583 = (2 * 3791) + 1 ;
hence not 2 divides 7583 by NAT_4:9; :: thesis: ( not 3 divides 7583 & not 5 divides 7583 & not 7 divides 7583 & not 11 divides 7583 & not 13 divides 7583 & not 17 divides 7583 & not 19 divides 7583 & not 23 divides 7583 & not 29 divides 7583 & not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (3 * 2527) + 2 ;
hence not 3 divides 7583 by NAT_4:9; :: thesis: ( not 5 divides 7583 & not 7 divides 7583 & not 11 divides 7583 & not 13 divides 7583 & not 17 divides 7583 & not 19 divides 7583 & not 23 divides 7583 & not 29 divides 7583 & not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (5 * 1516) + 3 ;
hence not 5 divides 7583 by NAT_4:9; :: thesis: ( not 7 divides 7583 & not 11 divides 7583 & not 13 divides 7583 & not 17 divides 7583 & not 19 divides 7583 & not 23 divides 7583 & not 29 divides 7583 & not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (7 * 1083) + 2 ;
hence not 7 divides 7583 by NAT_4:9; :: thesis: ( not 11 divides 7583 & not 13 divides 7583 & not 17 divides 7583 & not 19 divides 7583 & not 23 divides 7583 & not 29 divides 7583 & not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (11 * 689) + 4 ;
hence not 11 divides 7583 by NAT_4:9; :: thesis: ( not 13 divides 7583 & not 17 divides 7583 & not 19 divides 7583 & not 23 divides 7583 & not 29 divides 7583 & not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (13 * 583) + 4 ;
hence not 13 divides 7583 by NAT_4:9; :: thesis: ( not 17 divides 7583 & not 19 divides 7583 & not 23 divides 7583 & not 29 divides 7583 & not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (17 * 446) + 1 ;
hence not 17 divides 7583 by NAT_4:9; :: thesis: ( not 19 divides 7583 & not 23 divides 7583 & not 29 divides 7583 & not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (19 * 399) + 2 ;
hence not 19 divides 7583 by NAT_4:9; :: thesis: ( not 23 divides 7583 & not 29 divides 7583 & not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (23 * 329) + 16 ;
hence not 23 divides 7583 by NAT_4:9; :: thesis: ( not 29 divides 7583 & not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (29 * 261) + 14 ;
hence not 29 divides 7583 by NAT_4:9; :: thesis: ( not 31 divides 7583 & not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (31 * 244) + 19 ;
hence not 31 divides 7583 by NAT_4:9; :: thesis: ( not 37 divides 7583 & not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (37 * 204) + 35 ;
hence not 37 divides 7583 by NAT_4:9; :: thesis: ( not 41 divides 7583 & not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (41 * 184) + 39 ;
hence not 41 divides 7583 by NAT_4:9; :: thesis: ( not 43 divides 7583 & not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (43 * 176) + 15 ;
hence not 43 divides 7583 by NAT_4:9; :: thesis: ( not 47 divides 7583 & not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (47 * 161) + 16 ;
hence not 47 divides 7583 by NAT_4:9; :: thesis: ( not 53 divides 7583 & not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (53 * 143) + 4 ;
hence not 53 divides 7583 by NAT_4:9; :: thesis: ( not 59 divides 7583 & not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (59 * 128) + 31 ;
hence not 59 divides 7583 by NAT_4:9; :: thesis: ( not 61 divides 7583 & not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (61 * 124) + 19 ;
hence not 61 divides 7583 by NAT_4:9; :: thesis: ( not 67 divides 7583 & not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (67 * 113) + 12 ;
hence not 67 divides 7583 by NAT_4:9; :: thesis: ( not 71 divides 7583 & not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (71 * 106) + 57 ;
hence not 71 divides 7583 by NAT_4:9; :: thesis: ( not 73 divides 7583 & not 79 divides 7583 & not 83 divides 7583 )
7583 = (73 * 103) + 64 ;
hence not 73 divides 7583 by NAT_4:9; :: thesis: ( not 79 divides 7583 & not 83 divides 7583 )
7583 = (79 * 95) + 78 ;
hence not 79 divides 7583 by NAT_4:9; :: thesis: not 83 divides 7583
7583 = (83 * 91) + 30 ;
hence not 83 divides 7583 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 7583 & n is prime holds
not n divides 7583 by XPRIMET1:46;
hence 7583 is prime by NAT_4:14; :: thesis: verum