now :: thesis: ( not 2 divides 941 & not 3 divides 941 & not 5 divides 941 & not 7 divides 941 & not 11 divides 941 & not 13 divides 941 & not 17 divides 941 & not 19 divides 941 & not 23 divides 941 & not 29 divides 941 )

941 = (2 * 470) + 1 ;
hence not 2 divides 941 by NAT_4:9; :: thesis: ( not 3 divides 941 & not 5 divides 941 & not 7 divides 941 & not 11 divides 941 & not 13 divides 941 & not 17 divides 941 & not 19 divides 941 & not 23 divides 941 & not 29 divides 941 )
941 = (3 * 313) + 2 ;
hence not 3 divides 941 by NAT_4:9; :: thesis: ( not 5 divides 941 & not 7 divides 941 & not 11 divides 941 & not 13 divides 941 & not 17 divides 941 & not 19 divides 941 & not 23 divides 941 & not 29 divides 941 )
941 = (5 * 188) + 1 ;
hence not 5 divides 941 by NAT_4:9; :: thesis: ( not 7 divides 941 & not 11 divides 941 & not 13 divides 941 & not 17 divides 941 & not 19 divides 941 & not 23 divides 941 & not 29 divides 941 )
941 = (7 * 134) + 3 ;
hence not 7 divides 941 by NAT_4:9; :: thesis: ( not 11 divides 941 & not 13 divides 941 & not 17 divides 941 & not 19 divides 941 & not 23 divides 941 & not 29 divides 941 )
941 = (11 * 85) + 6 ;
hence not 11 divides 941 by NAT_4:9; :: thesis: ( not 13 divides 941 & not 17 divides 941 & not 19 divides 941 & not 23 divides 941 & not 29 divides 941 )
941 = (13 * 72) + 5 ;
hence not 13 divides 941 by NAT_4:9; :: thesis: ( not 17 divides 941 & not 19 divides 941 & not 23 divides 941 & not 29 divides 941 )
941 = (17 * 55) + 6 ;
hence not 17 divides 941 by NAT_4:9; :: thesis: ( not 19 divides 941 & not 23 divides 941 & not 29 divides 941 )
941 = (19 * 49) + 10 ;
hence not 19 divides 941 by NAT_4:9; :: thesis: ( not 23 divides 941 & not 29 divides 941 )
941 = (23 * 40) + 21 ;
hence not 23 divides 941 by NAT_4:9; :: thesis: not 29 divides 941
941 = (29 * 32) + 13 ;
hence not 29 divides 941 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 941 & n is prime holds
not n divides 941 by XPRIMET1:20;
hence 941 is prime by NAT_4:14; :: thesis: verum