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 Th27:
  p < 47 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 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43
  proof
    assume p < 47;
    then 1+1 < p+1 & p < 46+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p & p < 43;
      hence thesis by Th25;
    end;
    suppose 43 <= p & p <= 43+1;
      hence thesis by XPRIMES0:44,NAT_1:9;
    end;
    suppose 44 <= p & p <= 44+1;
      hence thesis by XPRIMES0:44,45,NAT_1:9;
    end;
    suppose 45 <= p & p <= 45+1;
      hence thesis by XPRIMES0:45,46,NAT_1:9;
    end;
  end;
