now :: thesis: ( not 2 divides 4951 & not 3 divides 4951 & not 5 divides 4951 & not 7 divides 4951 & not 11 divides 4951 & not 13 divides 4951 & not 17 divides 4951 & not 19 divides 4951 & not 23 divides 4951 & not 29 divides 4951 & not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )

4951 = (2 * 2475) + 1 ;
hence not 2 divides 4951 by NAT_4:9; :: thesis: ( not 3 divides 4951 & not 5 divides 4951 & not 7 divides 4951 & not 11 divides 4951 & not 13 divides 4951 & not 17 divides 4951 & not 19 divides 4951 & not 23 divides 4951 & not 29 divides 4951 & not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (3 * 1650) + 1 ;
hence not 3 divides 4951 by NAT_4:9; :: thesis: ( not 5 divides 4951 & not 7 divides 4951 & not 11 divides 4951 & not 13 divides 4951 & not 17 divides 4951 & not 19 divides 4951 & not 23 divides 4951 & not 29 divides 4951 & not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (5 * 990) + 1 ;
hence not 5 divides 4951 by NAT_4:9; :: thesis: ( not 7 divides 4951 & not 11 divides 4951 & not 13 divides 4951 & not 17 divides 4951 & not 19 divides 4951 & not 23 divides 4951 & not 29 divides 4951 & not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (7 * 707) + 2 ;
hence not 7 divides 4951 by NAT_4:9; :: thesis: ( not 11 divides 4951 & not 13 divides 4951 & not 17 divides 4951 & not 19 divides 4951 & not 23 divides 4951 & not 29 divides 4951 & not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (11 * 450) + 1 ;
hence not 11 divides 4951 by NAT_4:9; :: thesis: ( not 13 divides 4951 & not 17 divides 4951 & not 19 divides 4951 & not 23 divides 4951 & not 29 divides 4951 & not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (13 * 380) + 11 ;
hence not 13 divides 4951 by NAT_4:9; :: thesis: ( not 17 divides 4951 & not 19 divides 4951 & not 23 divides 4951 & not 29 divides 4951 & not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (17 * 291) + 4 ;
hence not 17 divides 4951 by NAT_4:9; :: thesis: ( not 19 divides 4951 & not 23 divides 4951 & not 29 divides 4951 & not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (19 * 260) + 11 ;
hence not 19 divides 4951 by NAT_4:9; :: thesis: ( not 23 divides 4951 & not 29 divides 4951 & not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (23 * 215) + 6 ;
hence not 23 divides 4951 by NAT_4:9; :: thesis: ( not 29 divides 4951 & not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (29 * 170) + 21 ;
hence not 29 divides 4951 by NAT_4:9; :: thesis: ( not 31 divides 4951 & not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (31 * 159) + 22 ;
hence not 31 divides 4951 by NAT_4:9; :: thesis: ( not 37 divides 4951 & not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (37 * 133) + 30 ;
hence not 37 divides 4951 by NAT_4:9; :: thesis: ( not 41 divides 4951 & not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (41 * 120) + 31 ;
hence not 41 divides 4951 by NAT_4:9; :: thesis: ( not 43 divides 4951 & not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (43 * 115) + 6 ;
hence not 43 divides 4951 by NAT_4:9; :: thesis: ( not 47 divides 4951 & not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (47 * 105) + 16 ;
hence not 47 divides 4951 by NAT_4:9; :: thesis: ( not 53 divides 4951 & not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (53 * 93) + 22 ;
hence not 53 divides 4951 by NAT_4:9; :: thesis: ( not 59 divides 4951 & not 61 divides 4951 & not 67 divides 4951 )
4951 = (59 * 83) + 54 ;
hence not 59 divides 4951 by NAT_4:9; :: thesis: ( not 61 divides 4951 & not 67 divides 4951 )
4951 = (61 * 81) + 10 ;
hence not 61 divides 4951 by NAT_4:9; :: thesis: not 67 divides 4951
4951 = (67 * 73) + 60 ;
hence not 67 divides 4951 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 4951 & n is prime holds
not n divides 4951 by XPRIMET1:38;
hence 4951 is prime by NAT_4:14; :: thesis: verum