let p be Prime; :: thesis: ( not p < 37 or p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 )
assume p < 37 ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 )
then ( 1 + 1 < p + 1 & p < 36 + 1 ) by XREAL_1:6, INT_2:def 4;

per cases then ( ( 2 <= p & p < 31 ) or ( 31 <= p & p <= 31 + 1 ) or ( 32 <= p & p <= 32 + 1 ) or ( 33 <= p & p <= 33 + 1 ) or ( 34 <= p & p <= 34 + 1 ) or ( 35 <= p & p <= 35 + 1 ) ) by NAT_1:13;

suppose ( 2 <= p & p < 31 ) ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 )

hence ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 ) by Ttool31a; :: thesis: verum

end;

suppose ( ( 31 <= p & p <= 31 + 1 ) or ( 32 <= p & p <= 32 + 1 ) or ( 33 <= p & p <= 33 + 1 ) or ( 34 <= p & p <= 34 + 1 ) or ( 35 <= p & p <= 35 + 1 ) ) ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 )

then p = 31 by XPRIMES0:32, XPRIMES0:33, XPRIMES0:34, XPRIMES0:35, XPRIMES0:36, NAT_1:9;
hence ( p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or p = 23 or p = 29 or p = 31 ) ; :: thesis: verum

end;

end;