now :: thesis: ( not 2 divides 761 & not 3 divides 761 & not 5 divides 761 & not 7 divides 761 & not 11 divides 761 & not 13 divides 761 & not 17 divides 761 & not 19 divides 761 & not 23 divides 761 )

761 = (2 * 380) + 1 ;
hence not 2 divides 761 by NAT_4:9; :: thesis: ( not 3 divides 761 & not 5 divides 761 & not 7 divides 761 & not 11 divides 761 & not 13 divides 761 & not 17 divides 761 & not 19 divides 761 & not 23 divides 761 )
761 = (3 * 253) + 2 ;
hence not 3 divides 761 by NAT_4:9; :: thesis: ( not 5 divides 761 & not 7 divides 761 & not 11 divides 761 & not 13 divides 761 & not 17 divides 761 & not 19 divides 761 & not 23 divides 761 )
761 = (5 * 152) + 1 ;
hence not 5 divides 761 by NAT_4:9; :: thesis: ( not 7 divides 761 & not 11 divides 761 & not 13 divides 761 & not 17 divides 761 & not 19 divides 761 & not 23 divides 761 )
761 = (7 * 108) + 5 ;
hence not 7 divides 761 by NAT_4:9; :: thesis: ( not 11 divides 761 & not 13 divides 761 & not 17 divides 761 & not 19 divides 761 & not 23 divides 761 )
761 = (11 * 69) + 2 ;
hence not 11 divides 761 by NAT_4:9; :: thesis: ( not 13 divides 761 & not 17 divides 761 & not 19 divides 761 & not 23 divides 761 )
761 = (13 * 58) + 7 ;
hence not 13 divides 761 by NAT_4:9; :: thesis: ( not 17 divides 761 & not 19 divides 761 & not 23 divides 761 )
761 = (17 * 44) + 13 ;
hence not 17 divides 761 by NAT_4:9; :: thesis: ( not 19 divides 761 & not 23 divides 761 )
761 = (19 * 40) + 1 ;
hence not 19 divides 761 by NAT_4:9; :: thesis: not 23 divides 761
761 = (23 * 33) + 2 ;
hence not 23 divides 761 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 761 & n is prime holds
not n divides 761 by XPRIMET1:18;
hence 761 is prime by NAT_4:14; :: thesis: verum