now :: thesis: ( not 2 divides 373 & not 3 divides 373 & not 5 divides 373 & not 7 divides 373 & not 11 divides 373 & not 13 divides 373 & not 17 divides 373 & not 19 divides 373 )

373 = (2 * 186) + 1 ;
hence not 2 divides 373 by NAT_4:9; :: thesis: ( not 3 divides 373 & not 5 divides 373 & not 7 divides 373 & not 11 divides 373 & not 13 divides 373 & not 17 divides 373 & not 19 divides 373 )
373 = (3 * 124) + 1 ;
hence not 3 divides 373 by NAT_4:9; :: thesis: ( not 5 divides 373 & not 7 divides 373 & not 11 divides 373 & not 13 divides 373 & not 17 divides 373 & not 19 divides 373 )
373 = (5 * 74) + 3 ;
hence not 5 divides 373 by NAT_4:9; :: thesis: ( not 7 divides 373 & not 11 divides 373 & not 13 divides 373 & not 17 divides 373 & not 19 divides 373 )
373 = (7 * 53) + 2 ;
hence not 7 divides 373 by NAT_4:9; :: thesis: ( not 11 divides 373 & not 13 divides 373 & not 17 divides 373 & not 19 divides 373 )
373 = (11 * 33) + 10 ;
hence not 11 divides 373 by NAT_4:9; :: thesis: ( not 13 divides 373 & not 17 divides 373 & not 19 divides 373 )
373 = (13 * 28) + 9 ;
hence not 13 divides 373 by NAT_4:9; :: thesis: ( not 17 divides 373 & not 19 divides 373 )
373 = (17 * 21) + 16 ;
hence not 17 divides 373 by NAT_4:9; :: thesis: not 19 divides 373
373 = (19 * 19) + 12 ;
hence not 19 divides 373 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 373 & n is prime holds
not n divides 373 by XPRIMET1:16;
hence 373 is prime by NAT_4:14; :: thesis: verum