reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;
reserve p for Prime;

theorem Th17:
  p < 29 implies
  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
  proof
    assume p < 29;
    then 1+1 < p+1 & p < 28+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p & p < 23;
      hence thesis by Th15;
    end;
    suppose 23 <= p & p <= 23+1;
      hence thesis by XPRIMES0:24,NAT_1:9;
    end;
    suppose 24 <= p & p <= 24+1;
      hence thesis by XPRIMES0:24,25,NAT_1:9;
    end;
    suppose 25 <= p & p <= 25+1;
      hence thesis by XPRIMES0:25,26,NAT_1:9;
    end;
    suppose 26 <= p & p <= 26+1;
      hence thesis by XPRIMES0:26,27,NAT_1:9;
    end;
    suppose 27 <= p & p <= 27+1;
      hence thesis by XPRIMES0:27,28,NAT_1:9;
    end;
  end;
