now :: thesis: ( not 2 divides 3727 & not 3 divides 3727 & not 5 divides 3727 & not 7 divides 3727 & not 11 divides 3727 & not 13 divides 3727 & not 17 divides 3727 & not 19 divides 3727 & not 23 divides 3727 & not 29 divides 3727 & not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )

3727 = (2 * 1863) + 1 ;
hence not 2 divides 3727 by NAT_4:9; :: thesis: ( not 3 divides 3727 & not 5 divides 3727 & not 7 divides 3727 & not 11 divides 3727 & not 13 divides 3727 & not 17 divides 3727 & not 19 divides 3727 & not 23 divides 3727 & not 29 divides 3727 & not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (3 * 1242) + 1 ;
hence not 3 divides 3727 by NAT_4:9; :: thesis: ( not 5 divides 3727 & not 7 divides 3727 & not 11 divides 3727 & not 13 divides 3727 & not 17 divides 3727 & not 19 divides 3727 & not 23 divides 3727 & not 29 divides 3727 & not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (5 * 745) + 2 ;
hence not 5 divides 3727 by NAT_4:9; :: thesis: ( not 7 divides 3727 & not 11 divides 3727 & not 13 divides 3727 & not 17 divides 3727 & not 19 divides 3727 & not 23 divides 3727 & not 29 divides 3727 & not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (7 * 532) + 3 ;
hence not 7 divides 3727 by NAT_4:9; :: thesis: ( not 11 divides 3727 & not 13 divides 3727 & not 17 divides 3727 & not 19 divides 3727 & not 23 divides 3727 & not 29 divides 3727 & not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (11 * 338) + 9 ;
hence not 11 divides 3727 by NAT_4:9; :: thesis: ( not 13 divides 3727 & not 17 divides 3727 & not 19 divides 3727 & not 23 divides 3727 & not 29 divides 3727 & not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (13 * 286) + 9 ;
hence not 13 divides 3727 by NAT_4:9; :: thesis: ( not 17 divides 3727 & not 19 divides 3727 & not 23 divides 3727 & not 29 divides 3727 & not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (17 * 219) + 4 ;
hence not 17 divides 3727 by NAT_4:9; :: thesis: ( not 19 divides 3727 & not 23 divides 3727 & not 29 divides 3727 & not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (19 * 196) + 3 ;
hence not 19 divides 3727 by NAT_4:9; :: thesis: ( not 23 divides 3727 & not 29 divides 3727 & not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (23 * 162) + 1 ;
hence not 23 divides 3727 by NAT_4:9; :: thesis: ( not 29 divides 3727 & not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (29 * 128) + 15 ;
hence not 29 divides 3727 by NAT_4:9; :: thesis: ( not 31 divides 3727 & not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (31 * 120) + 7 ;
hence not 31 divides 3727 by NAT_4:9; :: thesis: ( not 37 divides 3727 & not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (37 * 100) + 27 ;
hence not 37 divides 3727 by NAT_4:9; :: thesis: ( not 41 divides 3727 & not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (41 * 90) + 37 ;
hence not 41 divides 3727 by NAT_4:9; :: thesis: ( not 43 divides 3727 & not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (43 * 86) + 29 ;
hence not 43 divides 3727 by NAT_4:9; :: thesis: ( not 47 divides 3727 & not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (47 * 79) + 14 ;
hence not 47 divides 3727 by NAT_4:9; :: thesis: ( not 53 divides 3727 & not 59 divides 3727 & not 61 divides 3727 )
3727 = (53 * 70) + 17 ;
hence not 53 divides 3727 by NAT_4:9; :: thesis: ( not 59 divides 3727 & not 61 divides 3727 )
3727 = (59 * 63) + 10 ;
hence not 59 divides 3727 by NAT_4:9; :: thesis: not 61 divides 3727
3727 = (61 * 61) + 6 ;
hence not 61 divides 3727 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 3727 & n is prime holds
not n divides 3727 by XPRIMET1:36;
hence 3727 is prime by NAT_4:14; :: thesis: verum