now :: thesis: ( not 2 divides 3803 & not 3 divides 3803 & not 5 divides 3803 & not 7 divides 3803 & not 11 divides 3803 & not 13 divides 3803 & not 17 divides 3803 & not 19 divides 3803 & not 23 divides 3803 & not 29 divides 3803 & not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )

3803 = (2 * 1901) + 1 ;
hence not 2 divides 3803 by NAT_4:9; :: thesis: ( not 3 divides 3803 & not 5 divides 3803 & not 7 divides 3803 & not 11 divides 3803 & not 13 divides 3803 & not 17 divides 3803 & not 19 divides 3803 & not 23 divides 3803 & not 29 divides 3803 & not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (3 * 1267) + 2 ;
hence not 3 divides 3803 by NAT_4:9; :: thesis: ( not 5 divides 3803 & not 7 divides 3803 & not 11 divides 3803 & not 13 divides 3803 & not 17 divides 3803 & not 19 divides 3803 & not 23 divides 3803 & not 29 divides 3803 & not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (5 * 760) + 3 ;
hence not 5 divides 3803 by NAT_4:9; :: thesis: ( not 7 divides 3803 & not 11 divides 3803 & not 13 divides 3803 & not 17 divides 3803 & not 19 divides 3803 & not 23 divides 3803 & not 29 divides 3803 & not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (7 * 543) + 2 ;
hence not 7 divides 3803 by NAT_4:9; :: thesis: ( not 11 divides 3803 & not 13 divides 3803 & not 17 divides 3803 & not 19 divides 3803 & not 23 divides 3803 & not 29 divides 3803 & not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (11 * 345) + 8 ;
hence not 11 divides 3803 by NAT_4:9; :: thesis: ( not 13 divides 3803 & not 17 divides 3803 & not 19 divides 3803 & not 23 divides 3803 & not 29 divides 3803 & not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (13 * 292) + 7 ;
hence not 13 divides 3803 by NAT_4:9; :: thesis: ( not 17 divides 3803 & not 19 divides 3803 & not 23 divides 3803 & not 29 divides 3803 & not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (17 * 223) + 12 ;
hence not 17 divides 3803 by NAT_4:9; :: thesis: ( not 19 divides 3803 & not 23 divides 3803 & not 29 divides 3803 & not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (19 * 200) + 3 ;
hence not 19 divides 3803 by NAT_4:9; :: thesis: ( not 23 divides 3803 & not 29 divides 3803 & not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (23 * 165) + 8 ;
hence not 23 divides 3803 by NAT_4:9; :: thesis: ( not 29 divides 3803 & not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (29 * 131) + 4 ;
hence not 29 divides 3803 by NAT_4:9; :: thesis: ( not 31 divides 3803 & not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (31 * 122) + 21 ;
hence not 31 divides 3803 by NAT_4:9; :: thesis: ( not 37 divides 3803 & not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (37 * 102) + 29 ;
hence not 37 divides 3803 by NAT_4:9; :: thesis: ( not 41 divides 3803 & not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (41 * 92) + 31 ;
hence not 41 divides 3803 by NAT_4:9; :: thesis: ( not 43 divides 3803 & not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (43 * 88) + 19 ;
hence not 43 divides 3803 by NAT_4:9; :: thesis: ( not 47 divides 3803 & not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (47 * 80) + 43 ;
hence not 47 divides 3803 by NAT_4:9; :: thesis: ( not 53 divides 3803 & not 59 divides 3803 & not 61 divides 3803 )
3803 = (53 * 71) + 40 ;
hence not 53 divides 3803 by NAT_4:9; :: thesis: ( not 59 divides 3803 & not 61 divides 3803 )
3803 = (59 * 64) + 27 ;
hence not 59 divides 3803 by NAT_4:9; :: thesis: not 61 divides 3803
3803 = (61 * 62) + 21 ;
hence not 61 divides 3803 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 3803 & n is prime holds
not n divides 3803 by XPRIMET1:36;
hence 3803 is prime by NAT_4:14; :: thesis: verum