now :: thesis: ( not 2 divides 3659 & not 3 divides 3659 & not 5 divides 3659 & not 7 divides 3659 & not 11 divides 3659 & not 13 divides 3659 & not 17 divides 3659 & not 19 divides 3659 & not 23 divides 3659 & not 29 divides 3659 & not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )

3659 = (2 * 1829) + 1 ;
hence not 2 divides 3659 by NAT_4:9; :: thesis: ( not 3 divides 3659 & not 5 divides 3659 & not 7 divides 3659 & not 11 divides 3659 & not 13 divides 3659 & not 17 divides 3659 & not 19 divides 3659 & not 23 divides 3659 & not 29 divides 3659 & not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (3 * 1219) + 2 ;
hence not 3 divides 3659 by NAT_4:9; :: thesis: ( not 5 divides 3659 & not 7 divides 3659 & not 11 divides 3659 & not 13 divides 3659 & not 17 divides 3659 & not 19 divides 3659 & not 23 divides 3659 & not 29 divides 3659 & not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (5 * 731) + 4 ;
hence not 5 divides 3659 by NAT_4:9; :: thesis: ( not 7 divides 3659 & not 11 divides 3659 & not 13 divides 3659 & not 17 divides 3659 & not 19 divides 3659 & not 23 divides 3659 & not 29 divides 3659 & not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (7 * 522) + 5 ;
hence not 7 divides 3659 by NAT_4:9; :: thesis: ( not 11 divides 3659 & not 13 divides 3659 & not 17 divides 3659 & not 19 divides 3659 & not 23 divides 3659 & not 29 divides 3659 & not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (11 * 332) + 7 ;
hence not 11 divides 3659 by NAT_4:9; :: thesis: ( not 13 divides 3659 & not 17 divides 3659 & not 19 divides 3659 & not 23 divides 3659 & not 29 divides 3659 & not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (13 * 281) + 6 ;
hence not 13 divides 3659 by NAT_4:9; :: thesis: ( not 17 divides 3659 & not 19 divides 3659 & not 23 divides 3659 & not 29 divides 3659 & not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (17 * 215) + 4 ;
hence not 17 divides 3659 by NAT_4:9; :: thesis: ( not 19 divides 3659 & not 23 divides 3659 & not 29 divides 3659 & not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (19 * 192) + 11 ;
hence not 19 divides 3659 by NAT_4:9; :: thesis: ( not 23 divides 3659 & not 29 divides 3659 & not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (23 * 159) + 2 ;
hence not 23 divides 3659 by NAT_4:9; :: thesis: ( not 29 divides 3659 & not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (29 * 126) + 5 ;
hence not 29 divides 3659 by NAT_4:9; :: thesis: ( not 31 divides 3659 & not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (31 * 118) + 1 ;
hence not 31 divides 3659 by NAT_4:9; :: thesis: ( not 37 divides 3659 & not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (37 * 98) + 33 ;
hence not 37 divides 3659 by NAT_4:9; :: thesis: ( not 41 divides 3659 & not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (41 * 89) + 10 ;
hence not 41 divides 3659 by NAT_4:9; :: thesis: ( not 43 divides 3659 & not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (43 * 85) + 4 ;
hence not 43 divides 3659 by NAT_4:9; :: thesis: ( not 47 divides 3659 & not 53 divides 3659 & not 59 divides 3659 )
3659 = (47 * 77) + 40 ;
hence not 47 divides 3659 by NAT_4:9; :: thesis: ( not 53 divides 3659 & not 59 divides 3659 )
3659 = (53 * 69) + 2 ;
hence not 53 divides 3659 by NAT_4:9; :: thesis: not 59 divides 3659
3659 = (59 * 62) + 1 ;
hence not 59 divides 3659 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 3659 & n is prime holds
not n divides 3659 by XPRIMET1:34;
hence 3659 is prime by NAT_4:14; :: thesis: verum