now :: thesis: ( not 2 divides 787 & not 3 divides 787 & not 5 divides 787 & not 7 divides 787 & not 11 divides 787 & not 13 divides 787 & not 17 divides 787 & not 19 divides 787 & not 23 divides 787 )

787 = (2 * 393) + 1 ;
hence not 2 divides 787 by NAT_4:9; :: thesis: ( not 3 divides 787 & not 5 divides 787 & not 7 divides 787 & not 11 divides 787 & not 13 divides 787 & not 17 divides 787 & not 19 divides 787 & not 23 divides 787 )
787 = (3 * 262) + 1 ;
hence not 3 divides 787 by NAT_4:9; :: thesis: ( not 5 divides 787 & not 7 divides 787 & not 11 divides 787 & not 13 divides 787 & not 17 divides 787 & not 19 divides 787 & not 23 divides 787 )
787 = (5 * 157) + 2 ;
hence not 5 divides 787 by NAT_4:9; :: thesis: ( not 7 divides 787 & not 11 divides 787 & not 13 divides 787 & not 17 divides 787 & not 19 divides 787 & not 23 divides 787 )
787 = (7 * 112) + 3 ;
hence not 7 divides 787 by NAT_4:9; :: thesis: ( not 11 divides 787 & not 13 divides 787 & not 17 divides 787 & not 19 divides 787 & not 23 divides 787 )
787 = (11 * 71) + 6 ;
hence not 11 divides 787 by NAT_4:9; :: thesis: ( not 13 divides 787 & not 17 divides 787 & not 19 divides 787 & not 23 divides 787 )
787 = (13 * 60) + 7 ;
hence not 13 divides 787 by NAT_4:9; :: thesis: ( not 17 divides 787 & not 19 divides 787 & not 23 divides 787 )
787 = (17 * 46) + 5 ;
hence not 17 divides 787 by NAT_4:9; :: thesis: ( not 19 divides 787 & not 23 divides 787 )
787 = (19 * 41) + 8 ;
hence not 19 divides 787 by NAT_4:9; :: thesis: not 23 divides 787
787 = (23 * 34) + 5 ;
hence not 23 divides 787 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 787 & n is prime holds
not n divides 787 by XPRIMET1:18;
hence 787 is prime by NAT_4:14; :: thesis: verum