let p be Prime; :: thesis: ( not p < 19 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 )
assume p < 19 ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 )
then ( 1 + 1 < p + 1 & p < 18 + 1 ) by XREAL_1:6, INT_2:def 4;

per cases then ( ( 2 <= p & p < 17 ) or ( 17 <= p & p <= 17 + 1 ) ) by NAT_1:13;

suppose ( 2 <= p & p < 17 ) ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 )

hence ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 ) by Ttool17a; :: thesis: verum

end;

suppose ( 17 <= p & p <= 17 + 1 ) ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 )

hence ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 ) by XPRIMES0:18, NAT_1:9; :: thesis: verum

end;

end;