let p be Prime; :: thesis: ( not p < 167 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 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 )
assume p < 167 ; :: 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 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 )
then ( 1 + 1 < p + 1 & p < 166 + 1 ) by XREAL_1:6, INT_2:def 4;

per cases then ( ( 2 <= p & p < 163 ) or ( 163 <= p & p <= 163 + 1 ) or ( 164 <= p & p <= 164 + 1 ) or ( 165 <= p & p <= 165 + 1 ) ) by NAT_1:13;

suppose ( 2 <= p & p < 163 ) ; :: 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 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 )

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 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 ) by Ttool163a; :: thesis: verum

end;

suppose ( ( 163 <= p & p <= 163 + 1 ) or ( 164 <= p & p <= 164 + 1 ) or ( 165 <= p & p <= 165 + 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 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 )

then p = 163 by XPRIMES0:164, XPRIMES0:165, XPRIMES0:166, 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 or p = 41 or p = 43 or p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or p = 151 or p = 157 or p = 163 ) ; :: thesis: verum

end;

end;