let p be Prime; :: thesis: ( not p < 59 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 )
assume p < 59 ; :: 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 )
then ( 1 + 1 < p + 1 & p < 58 + 1 ) by XREAL_1:6, INT_2:def 4;

per cases then ( ( 2 <= p & p < 53 ) or ( 53 <= p & p <= 53 + 1 ) or ( 54 <= p & p <= 54 + 1 ) or ( 55 <= p & p <= 55 + 1 ) or ( 56 <= p & p <= 56 + 1 ) or ( 57 <= p & p <= 57 + 1 ) ) by NAT_1:13;

suppose ( 2 <= p & p < 53 ) ; :: 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 )

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 ) by Ttool53a; :: thesis: verum

end;

suppose ( ( 53 <= p & p <= 53 + 1 ) or ( 54 <= p & p <= 54 + 1 ) or ( 55 <= p & p <= 55 + 1 ) or ( 56 <= p & p <= 56 + 1 ) or ( 57 <= p & p <= 57 + 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 )

then p = 53 by XPRIMES0:54, XPRIMES0:55, XPRIMES0:56, XPRIMES0:57, XPRIMES0:58, 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 ) ; :: thesis: verum

end;

end;