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 Th23:
  p < 41 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
  proof
    assume p < 41;
    then 1+1 < p+1 & p < 40+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p & p < 37;
      hence thesis by Th21;
    end;
    suppose 37 <= p & p <= 37+1;
      hence thesis by XPRIMES0:38,NAT_1:9;
    end;
    suppose 38 <= p & p <= 38+1;
      hence thesis by XPRIMES0:38,39,NAT_1:9;
    end;
    suppose 39 <= p & p <= 39+1;
      hence thesis by XPRIMES0:39,40,NAT_1:9;
    end;
  end;
