now :: thesis: ( not 2 divides 757 & not 3 divides 757 & not 5 divides 757 & not 7 divides 757 & not 11 divides 757 & not 13 divides 757 & not 17 divides 757 & not 19 divides 757 & not 23 divides 757 )

757 = (2 * 378) + 1 ;
hence not 2 divides 757 by NAT_4:9; :: thesis: ( not 3 divides 757 & not 5 divides 757 & not 7 divides 757 & not 11 divides 757 & not 13 divides 757 & not 17 divides 757 & not 19 divides 757 & not 23 divides 757 )
757 = (3 * 252) + 1 ;
hence not 3 divides 757 by NAT_4:9; :: thesis: ( not 5 divides 757 & not 7 divides 757 & not 11 divides 757 & not 13 divides 757 & not 17 divides 757 & not 19 divides 757 & not 23 divides 757 )
757 = (5 * 151) + 2 ;
hence not 5 divides 757 by NAT_4:9; :: thesis: ( not 7 divides 757 & not 11 divides 757 & not 13 divides 757 & not 17 divides 757 & not 19 divides 757 & not 23 divides 757 )
757 = (7 * 108) + 1 ;
hence not 7 divides 757 by NAT_4:9; :: thesis: ( not 11 divides 757 & not 13 divides 757 & not 17 divides 757 & not 19 divides 757 & not 23 divides 757 )
757 = (11 * 68) + 9 ;
hence not 11 divides 757 by NAT_4:9; :: thesis: ( not 13 divides 757 & not 17 divides 757 & not 19 divides 757 & not 23 divides 757 )
757 = (13 * 58) + 3 ;
hence not 13 divides 757 by NAT_4:9; :: thesis: ( not 17 divides 757 & not 19 divides 757 & not 23 divides 757 )
757 = (17 * 44) + 9 ;
hence not 17 divides 757 by NAT_4:9; :: thesis: ( not 19 divides 757 & not 23 divides 757 )
757 = (19 * 39) + 16 ;
hence not 19 divides 757 by NAT_4:9; :: thesis: not 23 divides 757
757 = (23 * 32) + 21 ;
hence not 23 divides 757 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 757 & n is prime holds
not n divides 757 by XPRIMET1:18;
hence 757 is prime by NAT_4:14; :: thesis: verum