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 Th31:
  p < 59 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 or p = 47 or p = 53
  proof
    assume p < 59;
    then 1+1 < p+1 & p < 58+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p & p < 53;
      hence thesis by Th29;
    end;
    suppose 53 <= p & p <= 53+1;
      hence thesis by XPRIMES0:54,NAT_1:9;
    end;
    suppose 54 <= p & p <= 54+1;
      hence thesis by XPRIMES0:54,55,NAT_1:9;
    end;
    suppose 55 <= p & p <= 55+1;
      hence thesis by XPRIMES0:55,56,NAT_1:9;
    end;
    suppose 56 <= p & p <= 56+1;
      hence thesis by XPRIMES0:56,57,NAT_1:9;
    end;
    suppose 57 <= p & p <= 57+1;
      hence thesis by XPRIMES0:57,58,NAT_1:9;
    end;
  end;
