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

theorem
  for z being non zero Nat holds
  {k where k is Nat: k is odd & 2|^(2|^z)+k is composite} is infinite
  proof
    let z;
    set S = {k where k is Nat: k is odd & 2|^(2|^z)+k is composite};
    deffunc F(Nat) = 6*$1-1;
    consider f being ManySortedSet of NATPLUS such that
A1: for n being Element of NATPLUS holds f.n = F(n) from PBOOLE:sch 5;
    set R = rng f;
A2: dom f = NATPLUS by PARTFUN1:def 2;
A3: R c= S
    proof
      let a be object;
      assume a in R;
      then consider t being object such that
A4:   t in dom f and
A5:   a = f.t by FUNCT_1:def 3;
      reconsider t as Element of NATPLUS by A4,PARTFUN1:def 2;
A6:   a = 6*t-1 by A1,A5;
A7:   2*(3*t)-1 is odd;
      0+1 <= t by NAT_1:13;
      then 2|^(2|^z)+(6*t-1) is composite by Th64;
      hence a in S by A6,A7;
    end;
    for m being Nat ex n being Nat st n >= m & n in R
    proof
      let m be Nat;
      1*m <= 6*m by XREAL_1:64;
      then
A8:   m+0 <= 6*m+5 by XREAL_1:7;
      reconsider n = F(m+1) as Element of NAT by INT_1:3;
      take n;
      thus n >= m by A8;
A9:   m+1 in NATPLUS by NAT_LAT:def 6;
      then f.(m+1) = n by A1;
      hence n in R by A2,A9,FUNCT_1:def 3;
    end;
    hence thesis by A3,PYTHTRIP:9;
  end;
