let p be Prime; :: thesis: ( not p < 41 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 or p = 37 )
assume p < 41 ; :: 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 or p = 37 )
then ( 1 + 1 < p + 1 & p < 40 + 1 ) by XREAL_1:6, INT_2:def 4;

per cases then ( ( 2 <= p & p < 37 ) or ( 37 <= p & p <= 37 + 1 ) or ( 38 <= p & p <= 38 + 1 ) or ( 39 <= p & p <= 39 + 1 ) ) by NAT_1:13;

suppose ( 2 <= p & 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 or p = 37 )

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 or p = 37 ) by Ttool37a; :: thesis: verum

end;

suppose ( ( 37 <= p & p <= 37 + 1 ) or ( 38 <= p & p <= 38 + 1 ) or ( 39 <= p & p <= 39 + 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 or p = 37 )

then p = 37 by XPRIMES0:38, XPRIMES0:39, XPRIMES0:40, 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 or p = 37 ) ; :: thesis: verum

end;

end;