now :: thesis: ( not 2 divides 1951 & not 3 divides 1951 & not 5 divides 1951 & not 7 divides 1951 & not 11 divides 1951 & not 13 divides 1951 & not 17 divides 1951 & not 19 divides 1951 & not 23 divides 1951 & not 29 divides 1951 & not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )

1951 = (2 * 975) + 1 ;
hence not 2 divides 1951 by NAT_4:9; :: thesis: ( not 3 divides 1951 & not 5 divides 1951 & not 7 divides 1951 & not 11 divides 1951 & not 13 divides 1951 & not 17 divides 1951 & not 19 divides 1951 & not 23 divides 1951 & not 29 divides 1951 & not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (3 * 650) + 1 ;
hence not 3 divides 1951 by NAT_4:9; :: thesis: ( not 5 divides 1951 & not 7 divides 1951 & not 11 divides 1951 & not 13 divides 1951 & not 17 divides 1951 & not 19 divides 1951 & not 23 divides 1951 & not 29 divides 1951 & not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (5 * 390) + 1 ;
hence not 5 divides 1951 by NAT_4:9; :: thesis: ( not 7 divides 1951 & not 11 divides 1951 & not 13 divides 1951 & not 17 divides 1951 & not 19 divides 1951 & not 23 divides 1951 & not 29 divides 1951 & not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (7 * 278) + 5 ;
hence not 7 divides 1951 by NAT_4:9; :: thesis: ( not 11 divides 1951 & not 13 divides 1951 & not 17 divides 1951 & not 19 divides 1951 & not 23 divides 1951 & not 29 divides 1951 & not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (11 * 177) + 4 ;
hence not 11 divides 1951 by NAT_4:9; :: thesis: ( not 13 divides 1951 & not 17 divides 1951 & not 19 divides 1951 & not 23 divides 1951 & not 29 divides 1951 & not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (13 * 150) + 1 ;
hence not 13 divides 1951 by NAT_4:9; :: thesis: ( not 17 divides 1951 & not 19 divides 1951 & not 23 divides 1951 & not 29 divides 1951 & not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (17 * 114) + 13 ;
hence not 17 divides 1951 by NAT_4:9; :: thesis: ( not 19 divides 1951 & not 23 divides 1951 & not 29 divides 1951 & not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (19 * 102) + 13 ;
hence not 19 divides 1951 by NAT_4:9; :: thesis: ( not 23 divides 1951 & not 29 divides 1951 & not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (23 * 84) + 19 ;
hence not 23 divides 1951 by NAT_4:9; :: thesis: ( not 29 divides 1951 & not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (29 * 67) + 8 ;
hence not 29 divides 1951 by NAT_4:9; :: thesis: ( not 31 divides 1951 & not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (31 * 62) + 29 ;
hence not 31 divides 1951 by NAT_4:9; :: thesis: ( not 37 divides 1951 & not 41 divides 1951 & not 43 divides 1951 )
1951 = (37 * 52) + 27 ;
hence not 37 divides 1951 by NAT_4:9; :: thesis: ( not 41 divides 1951 & not 43 divides 1951 )
1951 = (41 * 47) + 24 ;
hence not 41 divides 1951 by NAT_4:9; :: thesis: not 43 divides 1951
1951 = (43 * 45) + 16 ;
hence not 43 divides 1951 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1951 & n is prime holds
not n divides 1951 by XPRIMET1:28;
hence 1951 is prime by NAT_4:14; :: thesis: verum