now :: thesis: ( not 2 divides 5791 & not 3 divides 5791 & not 5 divides 5791 & not 7 divides 5791 & not 11 divides 5791 & not 13 divides 5791 & not 17 divides 5791 & not 19 divides 5791 & not 23 divides 5791 & not 29 divides 5791 & not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )

5791 = (2 * 2895) + 1 ;
hence not 2 divides 5791 by NAT_4:9; :: thesis: ( not 3 divides 5791 & not 5 divides 5791 & not 7 divides 5791 & not 11 divides 5791 & not 13 divides 5791 & not 17 divides 5791 & not 19 divides 5791 & not 23 divides 5791 & not 29 divides 5791 & not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (3 * 1930) + 1 ;
hence not 3 divides 5791 by NAT_4:9; :: thesis: ( not 5 divides 5791 & not 7 divides 5791 & not 11 divides 5791 & not 13 divides 5791 & not 17 divides 5791 & not 19 divides 5791 & not 23 divides 5791 & not 29 divides 5791 & not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (5 * 1158) + 1 ;
hence not 5 divides 5791 by NAT_4:9; :: thesis: ( not 7 divides 5791 & not 11 divides 5791 & not 13 divides 5791 & not 17 divides 5791 & not 19 divides 5791 & not 23 divides 5791 & not 29 divides 5791 & not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (7 * 827) + 2 ;
hence not 7 divides 5791 by NAT_4:9; :: thesis: ( not 11 divides 5791 & not 13 divides 5791 & not 17 divides 5791 & not 19 divides 5791 & not 23 divides 5791 & not 29 divides 5791 & not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (11 * 526) + 5 ;
hence not 11 divides 5791 by NAT_4:9; :: thesis: ( not 13 divides 5791 & not 17 divides 5791 & not 19 divides 5791 & not 23 divides 5791 & not 29 divides 5791 & not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (13 * 445) + 6 ;
hence not 13 divides 5791 by NAT_4:9; :: thesis: ( not 17 divides 5791 & not 19 divides 5791 & not 23 divides 5791 & not 29 divides 5791 & not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (17 * 340) + 11 ;
hence not 17 divides 5791 by NAT_4:9; :: thesis: ( not 19 divides 5791 & not 23 divides 5791 & not 29 divides 5791 & not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (19 * 304) + 15 ;
hence not 19 divides 5791 by NAT_4:9; :: thesis: ( not 23 divides 5791 & not 29 divides 5791 & not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (23 * 251) + 18 ;
hence not 23 divides 5791 by NAT_4:9; :: thesis: ( not 29 divides 5791 & not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (29 * 199) + 20 ;
hence not 29 divides 5791 by NAT_4:9; :: thesis: ( not 31 divides 5791 & not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (31 * 186) + 25 ;
hence not 31 divides 5791 by NAT_4:9; :: thesis: ( not 37 divides 5791 & not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (37 * 156) + 19 ;
hence not 37 divides 5791 by NAT_4:9; :: thesis: ( not 41 divides 5791 & not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (41 * 141) + 10 ;
hence not 41 divides 5791 by NAT_4:9; :: thesis: ( not 43 divides 5791 & not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (43 * 134) + 29 ;
hence not 43 divides 5791 by NAT_4:9; :: thesis: ( not 47 divides 5791 & not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (47 * 123) + 10 ;
hence not 47 divides 5791 by NAT_4:9; :: thesis: ( not 53 divides 5791 & not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (53 * 109) + 14 ;
hence not 53 divides 5791 by NAT_4:9; :: thesis: ( not 59 divides 5791 & not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (59 * 98) + 9 ;
hence not 59 divides 5791 by NAT_4:9; :: thesis: ( not 61 divides 5791 & not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (61 * 94) + 57 ;
hence not 61 divides 5791 by NAT_4:9; :: thesis: ( not 67 divides 5791 & not 71 divides 5791 & not 73 divides 5791 )
5791 = (67 * 86) + 29 ;
hence not 67 divides 5791 by NAT_4:9; :: thesis: ( not 71 divides 5791 & not 73 divides 5791 )
5791 = (71 * 81) + 40 ;
hence not 71 divides 5791 by NAT_4:9; :: thesis: not 73 divides 5791
5791 = (73 * 79) + 24 ;
hence not 73 divides 5791 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 5791 & n is prime holds
not n divides 5791 by XPRIMET1:42;
hence 5791 is prime by NAT_4:14; :: thesis: verum