reserve k,n,m for Nat,
  a,x,X,Y for set,
  D,D1,D2,S for non empty set,
  p,q for FinSequence of NAT;
reserve F,F1,G,G1,H,H1,H2 for LTL-formula;

theorem Th2:
  H is atomic or H is negative or H is conjunctive or H is
  disjunctive or H is next or H is Until or H is Release
proof
A1: H is Element of LTL_WFF by Def10;
  assume
A2: not thesis;
  then atom.0 <> H;
  then
A3: not atom.0 in { H } by TARSKI:def 1;
A4: now
    let p,q;
    assume
A5: p in LTL_WFF \ { H } & q in LTL_WFF \ { H };
    then reconsider F = p, G = q as LTL-formula by Def10;
    F 'R' G <> H by A2;
    then
A6: not p 'R' q in { H } by TARSKI:def 1;
    p 'R' q in LTL_WFF by A5,Def9;
    hence p 'R' q in LTL_WFF \ { H } by A6,XBOOLE_0:def 5;
  end;
A7: now
    let p,q;
    assume
A8: p in LTL_WFF \ { H } & q in LTL_WFF \ { H };
    then reconsider F = p, G = q as LTL-formula by Def10;
    F 'U' G <> H by A2;
    then
A9: not p 'U' q in { H } by TARSKI:def 1;
    p 'U' q in LTL_WFF by A8,Def9;
    hence p 'U' q in LTL_WFF \ { H } by A9,XBOOLE_0:def 5;
  end;
A10: now
    let p;
    assume
A11: p in LTL_WFF \ { H };
    then reconsider H1 = p as LTL-formula by Def10;
    'X' H1 <> H by A2;
    then
A12: not 'X' p in { H } by TARSKI:def 1;
    'X' p in LTL_WFF by A11,Def9;
    hence 'X' p in LTL_WFF \ { H } by A12,XBOOLE_0:def 5;
  end;
A13: now
    let p,q;
    assume
A14: p in LTL_WFF \ { H } & q in LTL_WFF \ { H };
    then reconsider F = p, G = q as LTL-formula by Def10;
    F 'or' G <> H by A2;
    then
A15: not p 'or' q in { H } by TARSKI:def 1;
    p 'or' q in LTL_WFF by A14,Def9;
    hence p 'or' q in LTL_WFF \ { H } by A15,XBOOLE_0:def 5;
  end;
A16: now
    let p,q;
    assume
A17: p in LTL_WFF \ { H } & q in LTL_WFF \ { H };
    then reconsider F = p, G = q as LTL-formula by Def10;
    F '&' G <> H by A2;
    then
A18: not p '&' q in { H } by TARSKI:def 1;
    p '&' q in LTL_WFF by A17,Def9;
    hence p '&' q in LTL_WFF \ { H } by A18,XBOOLE_0:def 5;
  end;
A19: now
    let p;
    assume
A20: p in LTL_WFF \ { H };
    then reconsider H1 = p as LTL-formula by Def10;
    'not' H1 <> H by A2;
    then
A21: not 'not' p in { H } by TARSKI:def 1;
    'not' p in LTL_WFF by A20,Def9;
    hence 'not' p in LTL_WFF \ { H } by A21,XBOOLE_0:def 5;
  end;
A22: now
    let n;
    atom.n <> H by A2;
    then
A23: not atom.n in { H } by TARSKI:def 1;
    atom.n in LTL_WFF by Def9;
    hence atom.n in LTL_WFF \ { H } by A23,XBOOLE_0:def 5;
  end;
  atom.0 in LTL_WFF by Def9;
  then
A24: LTL_WFF \ { H } is non empty by A3,XBOOLE_0:def 5;
  for a st a in LTL_WFF \ { H } holds a is FinSequence of NAT by Def9;
  then LTL_WFF c= LTL_WFF \ { H } by A24,A22,A19,A16,A13,A10,A7,A4,Def9;
  then H in LTL_WFF \ { H } by A1;
  then not H in { H } by XBOOLE_0:def 5;
  hence contradiction by TARSKI:def 1;
end;
