let p be Prime; :: thesis: ( not p < 29 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 )
assume p < 29 ; :: 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 )
then ( 1 + 1 < p + 1 & p < 28 + 1 ) by XREAL_1:6, INT_2:def 4;
per cases then ( ( 2 <= p & p < 23 ) or ( 23 <= p & p <= 23 + 1 ) or ( 24 <= p & p <= 24 + 1 ) or ( 25 <= p & p <= 25 + 1 ) or ( 26 <= p & p <= 26 + 1 ) or ( 27 <= p & p <= 27 + 1 ) ) by NAT_1:13;
suppose ( 2 <= p & p < 23 ) ; :: 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 )
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 ) by Th15; :: thesis: verum
end;
suppose ( 23 <= p & p <= 23 + 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 )
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 ) by XPRIMES0:24, NAT_1:9; :: thesis: verum
end;
suppose ( 24 <= p & p <= 24 + 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 )
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 ) by XPRIMES0:24, XPRIMES0:25, NAT_1:9; :: thesis: verum
end;
suppose ( 25 <= p & p <= 25 + 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 )
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 ) by XPRIMES0:25, XPRIMES0:26, NAT_1:9; :: thesis: verum
end;
suppose ( 26 <= p & p <= 26 + 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 )
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 ) by XPRIMES0:26, XPRIMES0:27, NAT_1:9; :: thesis: verum
end;
suppose ( 27 <= p & p <= 27 + 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 )
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 ) by XPRIMES0:27, XPRIMES0:28, NAT_1:9; :: thesis: verum
end;
end;