let p be Prime; :: thesis: ( not p < 11 or p = 2 or p = 3 or p = 5 or p = 7 )
assume p < 11 ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 )
then ( 1 + 1 < p + 1 & p < 10 + 1 ) by XREAL_1:6, INT_2:def 4;

per cases then ( ( 2 <= p & p < 7 ) or ( 7 <= p & p <= 7 + 1 ) or ( 8 <= p & p <= 8 + 1 ) or ( 9 <= p & p <= 9 + 1 ) ) by NAT_1:13;

suppose ( 2 <= p & p < 7 ) ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 )

hence ( p = 2 or p = 3 or p = 5 or p = 7 ) by Th5; :: thesis: verum

end;

suppose ( 7 <= p & p <= 7 + 1 ) ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 )

hence ( p = 2 or p = 3 or p = 5 or p = 7 ) by XPRIMES0:8, NAT_1:9; :: thesis: verum

end;

suppose ( 8 <= p & p <= 8 + 1 ) ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 )

hence ( p = 2 or p = 3 or p = 5 or p = 7 ) by XPRIMES0:8, XPRIMES0:9, NAT_1:9; :: thesis: verum

end;

suppose ( 9 <= p & p <= 9 + 1 ) ; :: thesis: ( p = 2 or p = 3 or p = 5 or p = 7 )

hence ( p = 2 or p = 3 or p = 5 or p = 7 ) by XPRIMES0:9, XPRIMES0:10, NAT_1:9; :: thesis: verum

end;

end;