now :: thesis: ( not 2 divides 2753 & not 3 divides 2753 & not 5 divides 2753 & not 7 divides 2753 & not 11 divides 2753 & not 13 divides 2753 & not 17 divides 2753 & not 19 divides 2753 & not 23 divides 2753 & not 29 divides 2753 & not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )

2753 = (2 * 1376) + 1 ;
hence not 2 divides 2753 by NAT_4:9; :: thesis: ( not 3 divides 2753 & not 5 divides 2753 & not 7 divides 2753 & not 11 divides 2753 & not 13 divides 2753 & not 17 divides 2753 & not 19 divides 2753 & not 23 divides 2753 & not 29 divides 2753 & not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (3 * 917) + 2 ;
hence not 3 divides 2753 by NAT_4:9; :: thesis: ( not 5 divides 2753 & not 7 divides 2753 & not 11 divides 2753 & not 13 divides 2753 & not 17 divides 2753 & not 19 divides 2753 & not 23 divides 2753 & not 29 divides 2753 & not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (5 * 550) + 3 ;
hence not 5 divides 2753 by NAT_4:9; :: thesis: ( not 7 divides 2753 & not 11 divides 2753 & not 13 divides 2753 & not 17 divides 2753 & not 19 divides 2753 & not 23 divides 2753 & not 29 divides 2753 & not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (7 * 393) + 2 ;
hence not 7 divides 2753 by NAT_4:9; :: thesis: ( not 11 divides 2753 & not 13 divides 2753 & not 17 divides 2753 & not 19 divides 2753 & not 23 divides 2753 & not 29 divides 2753 & not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (11 * 250) + 3 ;
hence not 11 divides 2753 by NAT_4:9; :: thesis: ( not 13 divides 2753 & not 17 divides 2753 & not 19 divides 2753 & not 23 divides 2753 & not 29 divides 2753 & not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (13 * 211) + 10 ;
hence not 13 divides 2753 by NAT_4:9; :: thesis: ( not 17 divides 2753 & not 19 divides 2753 & not 23 divides 2753 & not 29 divides 2753 & not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (17 * 161) + 16 ;
hence not 17 divides 2753 by NAT_4:9; :: thesis: ( not 19 divides 2753 & not 23 divides 2753 & not 29 divides 2753 & not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (19 * 144) + 17 ;
hence not 19 divides 2753 by NAT_4:9; :: thesis: ( not 23 divides 2753 & not 29 divides 2753 & not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (23 * 119) + 16 ;
hence not 23 divides 2753 by NAT_4:9; :: thesis: ( not 29 divides 2753 & not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (29 * 94) + 27 ;
hence not 29 divides 2753 by NAT_4:9; :: thesis: ( not 31 divides 2753 & not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (31 * 88) + 25 ;
hence not 31 divides 2753 by NAT_4:9; :: thesis: ( not 37 divides 2753 & not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (37 * 74) + 15 ;
hence not 37 divides 2753 by NAT_4:9; :: thesis: ( not 41 divides 2753 & not 43 divides 2753 & not 47 divides 2753 )
2753 = (41 * 67) + 6 ;
hence not 41 divides 2753 by NAT_4:9; :: thesis: ( not 43 divides 2753 & not 47 divides 2753 )
2753 = (43 * 64) + 1 ;
hence not 43 divides 2753 by NAT_4:9; :: thesis: not 47 divides 2753
2753 = (47 * 58) + 27 ;
hence not 47 divides 2753 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 2753 & n is prime holds
not n divides 2753 by XPRIMET1:30;
hence 2753 is prime by NAT_4:14; :: thesis: verum