now :: thesis: ( not 2 divides 1913 & not 3 divides 1913 & not 5 divides 1913 & not 7 divides 1913 & not 11 divides 1913 & not 13 divides 1913 & not 17 divides 1913 & not 19 divides 1913 & not 23 divides 1913 & not 29 divides 1913 & not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )

1913 = (2 * 956) + 1 ;
hence not 2 divides 1913 by NAT_4:9; :: thesis: ( not 3 divides 1913 & not 5 divides 1913 & not 7 divides 1913 & not 11 divides 1913 & not 13 divides 1913 & not 17 divides 1913 & not 19 divides 1913 & not 23 divides 1913 & not 29 divides 1913 & not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (3 * 637) + 2 ;
hence not 3 divides 1913 by NAT_4:9; :: thesis: ( not 5 divides 1913 & not 7 divides 1913 & not 11 divides 1913 & not 13 divides 1913 & not 17 divides 1913 & not 19 divides 1913 & not 23 divides 1913 & not 29 divides 1913 & not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (5 * 382) + 3 ;
hence not 5 divides 1913 by NAT_4:9; :: thesis: ( not 7 divides 1913 & not 11 divides 1913 & not 13 divides 1913 & not 17 divides 1913 & not 19 divides 1913 & not 23 divides 1913 & not 29 divides 1913 & not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (7 * 273) + 2 ;
hence not 7 divides 1913 by NAT_4:9; :: thesis: ( not 11 divides 1913 & not 13 divides 1913 & not 17 divides 1913 & not 19 divides 1913 & not 23 divides 1913 & not 29 divides 1913 & not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (11 * 173) + 10 ;
hence not 11 divides 1913 by NAT_4:9; :: thesis: ( not 13 divides 1913 & not 17 divides 1913 & not 19 divides 1913 & not 23 divides 1913 & not 29 divides 1913 & not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (13 * 147) + 2 ;
hence not 13 divides 1913 by NAT_4:9; :: thesis: ( not 17 divides 1913 & not 19 divides 1913 & not 23 divides 1913 & not 29 divides 1913 & not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (17 * 112) + 9 ;
hence not 17 divides 1913 by NAT_4:9; :: thesis: ( not 19 divides 1913 & not 23 divides 1913 & not 29 divides 1913 & not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (19 * 100) + 13 ;
hence not 19 divides 1913 by NAT_4:9; :: thesis: ( not 23 divides 1913 & not 29 divides 1913 & not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (23 * 83) + 4 ;
hence not 23 divides 1913 by NAT_4:9; :: thesis: ( not 29 divides 1913 & not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (29 * 65) + 28 ;
hence not 29 divides 1913 by NAT_4:9; :: thesis: ( not 31 divides 1913 & not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (31 * 61) + 22 ;
hence not 31 divides 1913 by NAT_4:9; :: thesis: ( not 37 divides 1913 & not 41 divides 1913 & not 43 divides 1913 )
1913 = (37 * 51) + 26 ;
hence not 37 divides 1913 by NAT_4:9; :: thesis: ( not 41 divides 1913 & not 43 divides 1913 )
1913 = (41 * 46) + 27 ;
hence not 41 divides 1913 by NAT_4:9; :: thesis: not 43 divides 1913
1913 = (43 * 44) + 21 ;
hence not 43 divides 1913 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 1913 & n is prime holds
not n divides 1913 by XPRIMET1:28;
hence 1913 is prime by NAT_4:14; :: thesis: verum