now :: thesis: ( not 2 divides 7541 & not 3 divides 7541 & not 5 divides 7541 & not 7 divides 7541 & not 11 divides 7541 & not 13 divides 7541 & not 17 divides 7541 & not 19 divides 7541 & not 23 divides 7541 & not 29 divides 7541 & not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )

7541 = (2 * 3770) + 1 ;
hence not 2 divides 7541 by NAT_4:9; :: thesis: ( not 3 divides 7541 & not 5 divides 7541 & not 7 divides 7541 & not 11 divides 7541 & not 13 divides 7541 & not 17 divides 7541 & not 19 divides 7541 & not 23 divides 7541 & not 29 divides 7541 & not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (3 * 2513) + 2 ;
hence not 3 divides 7541 by NAT_4:9; :: thesis: ( not 5 divides 7541 & not 7 divides 7541 & not 11 divides 7541 & not 13 divides 7541 & not 17 divides 7541 & not 19 divides 7541 & not 23 divides 7541 & not 29 divides 7541 & not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (5 * 1508) + 1 ;
hence not 5 divides 7541 by NAT_4:9; :: thesis: ( not 7 divides 7541 & not 11 divides 7541 & not 13 divides 7541 & not 17 divides 7541 & not 19 divides 7541 & not 23 divides 7541 & not 29 divides 7541 & not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (7 * 1077) + 2 ;
hence not 7 divides 7541 by NAT_4:9; :: thesis: ( not 11 divides 7541 & not 13 divides 7541 & not 17 divides 7541 & not 19 divides 7541 & not 23 divides 7541 & not 29 divides 7541 & not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (11 * 685) + 6 ;
hence not 11 divides 7541 by NAT_4:9; :: thesis: ( not 13 divides 7541 & not 17 divides 7541 & not 19 divides 7541 & not 23 divides 7541 & not 29 divides 7541 & not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (13 * 580) + 1 ;
hence not 13 divides 7541 by NAT_4:9; :: thesis: ( not 17 divides 7541 & not 19 divides 7541 & not 23 divides 7541 & not 29 divides 7541 & not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (17 * 443) + 10 ;
hence not 17 divides 7541 by NAT_4:9; :: thesis: ( not 19 divides 7541 & not 23 divides 7541 & not 29 divides 7541 & not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (19 * 396) + 17 ;
hence not 19 divides 7541 by NAT_4:9; :: thesis: ( not 23 divides 7541 & not 29 divides 7541 & not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (23 * 327) + 20 ;
hence not 23 divides 7541 by NAT_4:9; :: thesis: ( not 29 divides 7541 & not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (29 * 260) + 1 ;
hence not 29 divides 7541 by NAT_4:9; :: thesis: ( not 31 divides 7541 & not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (31 * 243) + 8 ;
hence not 31 divides 7541 by NAT_4:9; :: thesis: ( not 37 divides 7541 & not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (37 * 203) + 30 ;
hence not 37 divides 7541 by NAT_4:9; :: thesis: ( not 41 divides 7541 & not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (41 * 183) + 38 ;
hence not 41 divides 7541 by NAT_4:9; :: thesis: ( not 43 divides 7541 & not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (43 * 175) + 16 ;
hence not 43 divides 7541 by NAT_4:9; :: thesis: ( not 47 divides 7541 & not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (47 * 160) + 21 ;
hence not 47 divides 7541 by NAT_4:9; :: thesis: ( not 53 divides 7541 & not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (53 * 142) + 15 ;
hence not 53 divides 7541 by NAT_4:9; :: thesis: ( not 59 divides 7541 & not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (59 * 127) + 48 ;
hence not 59 divides 7541 by NAT_4:9; :: thesis: ( not 61 divides 7541 & not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (61 * 123) + 38 ;
hence not 61 divides 7541 by NAT_4:9; :: thesis: ( not 67 divides 7541 & not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (67 * 112) + 37 ;
hence not 67 divides 7541 by NAT_4:9; :: thesis: ( not 71 divides 7541 & not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (71 * 106) + 15 ;
hence not 71 divides 7541 by NAT_4:9; :: thesis: ( not 73 divides 7541 & not 79 divides 7541 & not 83 divides 7541 )
7541 = (73 * 103) + 22 ;
hence not 73 divides 7541 by NAT_4:9; :: thesis: ( not 79 divides 7541 & not 83 divides 7541 )
7541 = (79 * 95) + 36 ;
hence not 79 divides 7541 by NAT_4:9; :: thesis: not 83 divides 7541
7541 = (83 * 90) + 71 ;
hence not 83 divides 7541 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 7541 & n is prime holds
not n divides 7541 by XPRIMET1:46;
hence 7541 is prime by NAT_4:14; :: thesis: verum