reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem Th11:
  n < 31 & n is prime implies
  n = 2 or n = 3 or n = 5 or n = 7 or n = 11 or n = 13 or n = 17 or n = 19 or
  n = 23 or n = 29
  proof
    assume that
A1: n < 31 and
A2: n is prime;
A3: 1+1 < n+1 & n < 30+1 by A2,A1,XREAL_1:6;
    per cases by A3,NAT_1:13;
    suppose
      2 <= n & n < 5;
      hence thesis by A2,NAT_4:59;
    end;
    suppose
A4:   5 <= n & n <= 29+1;
      per cases by A4;
      suppose
        5 <= n & n <= 9+1;
        then
        5<=n & n<=5+1 or 6<=n & n<=6+1 or 7<=n & n<=7+1 or 8<=n & n<=8+1 or
        9<=n & n<=9+1;
        hence thesis by A2,NAT_4:57,NAT_1:9;
      end;
      suppose
        10 <= n & n <= 15+1;
        then
        10<=n & n<=10+1 or 11<=n & n<=11+1 or 12<=n & n<=12+1 or
        13<=n & n<=13+1 or 14<=n & n<=14+1 or 15<=n & n<=15+1;
        hence thesis by A2,NAT_4:57,NAT_1:9;
      end;
      suppose
        16 <= n & n <= 20+1;
        then
        16<=n & n<=16+1 or 17<=n & n<=17+1 or 18<=n & n<=18+1 or
        19<=n & n<=19+1 or 20<=n & n<=20+1;
        hence thesis by A2,Lm1,Lm2,NAT_1:9;
      end;
      suppose
        21 <= n & n <= 27+1;
        then
        21<=n & n<=21+1 or 22<=n & n<=22+1 or 23<=n & n<=23+1 or
        24<=n & n<=24+1 or 25<=n & n<=25+1 or 26<=n & n<=26+1 or
        27<=n & n<=27+1;
        hence thesis by A2,NAT_4:58,NAT_1:9;
      end;
      suppose
        28 <= n & n <= 29+1;
        then 28<=n & n<=28+1 or 29<=n & n<=29+1;
        hence thesis by A2,NAT_4:58,Th10,NAT_1:9;
      end;
    end;
  end;
