 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem
  for n being non zero Nat
  for x being object st x in [:bool SetPrimes n, Seg n:] holds
    x`1 is finite Subset of SetPrimes
  proof
    let n be non zero Nat;
    let x be object;
    assume G1: x in [:bool SetPrimes n, Seg n:];
    x is pair by CARDFIL4:4,G1; then
    x = [x`1,x`2]; then
    reconsider g1 = x`1 as Subset of SetPrimes n by G1,ZFMISC_1:87;
    SetPrimes n c= SetPrimes by XBOOLE_1:17; then
    g1 c= SetPrimes;
    hence thesis;
  end;
