now :: thesis: ( not 2 divides 503 & not 3 divides 503 & not 5 divides 503 & not 7 divides 503 & not 11 divides 503 & not 13 divides 503 & not 17 divides 503 & not 19 divides 503 )

503 = (2 * 251) + 1 ;
hence not 2 divides 503 by NAT_4:9; :: thesis: ( not 3 divides 503 & not 5 divides 503 & not 7 divides 503 & not 11 divides 503 & not 13 divides 503 & not 17 divides 503 & not 19 divides 503 )
503 = (3 * 167) + 2 ;
hence not 3 divides 503 by NAT_4:9; :: thesis: ( not 5 divides 503 & not 7 divides 503 & not 11 divides 503 & not 13 divides 503 & not 17 divides 503 & not 19 divides 503 )
503 = (5 * 100) + 3 ;
hence not 5 divides 503 by NAT_4:9; :: thesis: ( not 7 divides 503 & not 11 divides 503 & not 13 divides 503 & not 17 divides 503 & not 19 divides 503 )
503 = (7 * 71) + 6 ;
hence not 7 divides 503 by NAT_4:9; :: thesis: ( not 11 divides 503 & not 13 divides 503 & not 17 divides 503 & not 19 divides 503 )
503 = (11 * 45) + 8 ;
hence not 11 divides 503 by NAT_4:9; :: thesis: ( not 13 divides 503 & not 17 divides 503 & not 19 divides 503 )
503 = (13 * 38) + 9 ;
hence not 13 divides 503 by NAT_4:9; :: thesis: ( not 17 divides 503 & not 19 divides 503 )
503 = (17 * 29) + 10 ;
hence not 17 divides 503 by NAT_4:9; :: thesis: not 19 divides 503
503 = (19 * 26) + 9 ;
hence not 19 divides 503 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 503 & n is prime holds
not n divides 503 by XPRIMET1:16;
hence 503 is prime by NAT_4:14; :: thesis: verum