reserve k,n for Element of NAT,
  a,Y for set,
  D,D1,D2 for non empty set,
  p,q for FinSequence of NAT;
reserve F,F1,G,G1,H,H1,H2 for CTL-formula;

theorem Th2:
  H is atomic or H is negative or H is conjunctive or H is
  ExistNext or H is ExistGlobal or H is ExistUntill
proof
A1: H is Element of CTL_WFF by Def13;
  assume
A2: not thesis;
  then atom.0 <> H;
  then
A3: not atom.0 in { H } by TARSKI:def 1;
A4: now
    let p;
    assume
A5: p in CTL_WFF \ { H };
    then reconsider H1 = p as CTL-formula by Def13;
    EG H1 <> H by A2;
    then
A6: not EG p in { H } by TARSKI:def 1;
    EG p in CTL_WFF by A5,Def12;
    hence EG p in CTL_WFF \ { H } by A6,XBOOLE_0:def 5;
  end;
A7: now
    let p;
    assume
A8: p in CTL_WFF \ { H };
    then reconsider H1 = p as CTL-formula by Def13;
    EX H1 <> H by A2;
    then
A9: not EX p in { H } by TARSKI:def 1;
    EX p in CTL_WFF by A8,Def12;
    hence EX p in CTL_WFF \ { H } by A9,XBOOLE_0:def 5;
  end;
A10: now
    let p,q;
    assume that
A11: p in CTL_WFF \ { H } and
A12: q in CTL_WFF \ { H };
    reconsider F = p, G = q as CTL-formula by A11,A12,Def13;
    F '&' G <> H by A2;
    then
A13: not p '&' q in { H } by TARSKI:def 1;
    p '&' q in CTL_WFF by A11,A12,Def12;
    hence p '&' q in CTL_WFF \ { H } by A13,XBOOLE_0:def 5;
  end;
A14: now
    let p;
    assume
A15: p in CTL_WFF \ { H };
    then reconsider H1 = p as CTL-formula by Def13;
    'not' H1 <> H by A2;
    then
A16: not 'not' p in { H } by TARSKI:def 1;
    'not' p in CTL_WFF by A15,Def12;
    hence 'not' p in CTL_WFF \ { H } by A16,XBOOLE_0:def 5;
  end;
A17: now
    let p,q;
    assume that
A18: p in CTL_WFF \ { H } and
A19: q in CTL_WFF \ { H };
    reconsider F = p, G = q as CTL-formula by A18,A19,Def13;
    F EU G <> H by A2;
    then
A20: not p EU q in { H } by TARSKI:def 1;
    p EU q in CTL_WFF by A18,A19,Def12;
    hence p EU q in CTL_WFF \ { H } by A20,XBOOLE_0:def 5;
  end;
A21: now
    let n;
    atom.n <> H by A2;
    then
A22: not atom.n in { H } by TARSKI:def 1;
    atom.n in CTL_WFF by Def12;
    hence atom.n in CTL_WFF \ { H } by A22,XBOOLE_0:def 5;
  end;
  atom.0 in CTL_WFF by Def12;
  then
A23: CTL_WFF \ { H } is non empty by A3,XBOOLE_0:def 5;
  for a st a in CTL_WFF \ { H } holds a is FinSequence of NAT by Def12;
  then CTL_WFF c= CTL_WFF \ { H } by A23,A21,A14,A10,A7,A4,A17,Def12;
  then H in CTL_WFF \ { H } by A1;
  then not H in { H } by XBOOLE_0:def 5;
  hence contradiction by TARSKI:def 1;
end;
