now :: thesis: ( not 2 divides 7963 & not 3 divides 7963 & not 5 divides 7963 & not 7 divides 7963 & not 11 divides 7963 & not 13 divides 7963 & not 17 divides 7963 & not 19 divides 7963 & not 23 divides 7963 & not 29 divides 7963 & not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )

7963 = (2 * 3981) + 1 ;
hence not 2 divides 7963 by NAT_4:9; :: thesis: ( not 3 divides 7963 & not 5 divides 7963 & not 7 divides 7963 & not 11 divides 7963 & not 13 divides 7963 & not 17 divides 7963 & not 19 divides 7963 & not 23 divides 7963 & not 29 divides 7963 & not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (3 * 2654) + 1 ;
hence not 3 divides 7963 by NAT_4:9; :: thesis: ( not 5 divides 7963 & not 7 divides 7963 & not 11 divides 7963 & not 13 divides 7963 & not 17 divides 7963 & not 19 divides 7963 & not 23 divides 7963 & not 29 divides 7963 & not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (5 * 1592) + 3 ;
hence not 5 divides 7963 by NAT_4:9; :: thesis: ( not 7 divides 7963 & not 11 divides 7963 & not 13 divides 7963 & not 17 divides 7963 & not 19 divides 7963 & not 23 divides 7963 & not 29 divides 7963 & not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (7 * 1137) + 4 ;
hence not 7 divides 7963 by NAT_4:9; :: thesis: ( not 11 divides 7963 & not 13 divides 7963 & not 17 divides 7963 & not 19 divides 7963 & not 23 divides 7963 & not 29 divides 7963 & not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (11 * 723) + 10 ;
hence not 11 divides 7963 by NAT_4:9; :: thesis: ( not 13 divides 7963 & not 17 divides 7963 & not 19 divides 7963 & not 23 divides 7963 & not 29 divides 7963 & not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (13 * 612) + 7 ;
hence not 13 divides 7963 by NAT_4:9; :: thesis: ( not 17 divides 7963 & not 19 divides 7963 & not 23 divides 7963 & not 29 divides 7963 & not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (17 * 468) + 7 ;
hence not 17 divides 7963 by NAT_4:9; :: thesis: ( not 19 divides 7963 & not 23 divides 7963 & not 29 divides 7963 & not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (19 * 419) + 2 ;
hence not 19 divides 7963 by NAT_4:9; :: thesis: ( not 23 divides 7963 & not 29 divides 7963 & not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (23 * 346) + 5 ;
hence not 23 divides 7963 by NAT_4:9; :: thesis: ( not 29 divides 7963 & not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (29 * 274) + 17 ;
hence not 29 divides 7963 by NAT_4:9; :: thesis: ( not 31 divides 7963 & not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (31 * 256) + 27 ;
hence not 31 divides 7963 by NAT_4:9; :: thesis: ( not 37 divides 7963 & not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (37 * 215) + 8 ;
hence not 37 divides 7963 by NAT_4:9; :: thesis: ( not 41 divides 7963 & not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (41 * 194) + 9 ;
hence not 41 divides 7963 by NAT_4:9; :: thesis: ( not 43 divides 7963 & not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (43 * 185) + 8 ;
hence not 43 divides 7963 by NAT_4:9; :: thesis: ( not 47 divides 7963 & not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (47 * 169) + 20 ;
hence not 47 divides 7963 by NAT_4:9; :: thesis: ( not 53 divides 7963 & not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (53 * 150) + 13 ;
hence not 53 divides 7963 by NAT_4:9; :: thesis: ( not 59 divides 7963 & not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (59 * 134) + 57 ;
hence not 59 divides 7963 by NAT_4:9; :: thesis: ( not 61 divides 7963 & not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (61 * 130) + 33 ;
hence not 61 divides 7963 by NAT_4:9; :: thesis: ( not 67 divides 7963 & not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (67 * 118) + 57 ;
hence not 67 divides 7963 by NAT_4:9; :: thesis: ( not 71 divides 7963 & not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (71 * 112) + 11 ;
hence not 71 divides 7963 by NAT_4:9; :: thesis: ( not 73 divides 7963 & not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (73 * 109) + 6 ;
hence not 73 divides 7963 by NAT_4:9; :: thesis: ( not 79 divides 7963 & not 83 divides 7963 & not 89 divides 7963 )
7963 = (79 * 100) + 63 ;
hence not 79 divides 7963 by NAT_4:9; :: thesis: ( not 83 divides 7963 & not 89 divides 7963 )
7963 = (83 * 95) + 78 ;
hence not 83 divides 7963 by NAT_4:9; :: thesis: not 89 divides 7963
7963 = (89 * 89) + 42 ;
hence not 89 divides 7963 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 7963 & n is prime holds
not n divides 7963 by XPRIMET1:48;
hence 7963 is prime by NAT_4:14; :: thesis: verum