reserve A,B,p,q,r for Element of LTLB_WFF,
  M for LTLModel,
  j,k,n for Element of NAT,
  i for Nat,
  X for Subset of LTLB_WFF,
  F for finite Subset of LTLB_WFF,
  f for FinSequence of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN,
  x,y,z for set,
  P,Q,R for PNPair;

theorem Th15: for F be non empty finite Subset of LTLB_WFF
  ex p st p in tau F & tau ((tau F) \ {p}) = (tau F) \ {p}
  proof
    let F be non empty finite Subset of l;
    set G = {A where A is Element of l: A in tau F & A is conditional};
A1: G c= tau F
    proof
      let x be object;
      assume x in G;
      then ex A st A = x & A in tau F & A is conditional;
      hence thesis;
    end;
A2: G is FinSequenceSet of NAT
    proof
      let x be object;
      assume x in G;
      then ex A st A = x & A in tau F & A is conditional;
      hence thesis by Th2;
    end;
    per cases;
    suppose
A3:   G = {};
A4:   F is without_implication
      proof
        assume not F is without_implication;
        then consider p such that
A5:     p in F & p is conditional;
        F c= tau F by LTLAXIO3:16;
        then p in G by A5;
        hence contradiction by A3;
      end;
      consider p be object such that
A6:   p in F by XBOOLE_0:def 1;
      reconsider p as Element of l by A6;
      set Fp = (tau F) \ {p};
      Fp c= tau F by XBOOLE_1:36;
      then A7: Fp c= F by LTLAXIO3:18,A4;
A8:   tau Fp c= Fp
      proof
        let x be object;
        assume x in tau Fp;
        then consider A such that
A9:     A in Fp and
A10:    x in tau1.A by LTLAXIO3:def 5;
        x in {A} by A7,A9,A4,A10,LTLAXIO3:5;
        hence thesis by TARSKI:def 1,A9;
      end;
      Fp c= tau Fp & p in tau F by LTLAXIO3:16, A4,A6,LTLAXIO3:18;
      hence thesis by A8,XBOOLE_0:def 10;
    end;
    suppose
      G <> {};
      then reconsider G as non empty finite FinSequenceSet of NAT by A2,A1;
      consider A be FinSequence such that
A11:  A in G and
A12:  for B be FinSequence st B in G holds len B <= len A by Th1;
      set Fp = (tau F) \ {A};
A13:  Fp c= tau F by XBOOLE_1:36;
A14:  tau Fp c= Fp
      proof
        let x be object;
        assume x in tau Fp;
        then consider p such that
A15:    p in Fp and
A16:    x in tau1.p by LTLAXIO3:def 5;
A17:    not p in {A} by XBOOLE_0:def 5,A15;
        then A18: p <> A by TARSKI:def 1;
        x <> A
        proof
          per cases;
          suppose
            p is conditional;
            then p in G by A15,A13;
            then A19: len A >= len p by A12;
            per cases;
            suppose x = p;
              hence thesis by A17,TARSKI:def 1;
            end;
            suppose x <> p;
              hence thesis by LTLAXIO3:11,A16,A19;
            end;
          end;
          suppose not p is conditional;
            then x in {p} by LTLAXIO3:5,A16;
            hence thesis by TARSKI:def 1,A18;
          end;
        end;
        then A20: not x in {A} by TARSKI:def 1;
        tau1.p c= tau F by LTLAXIO3:23, A15,A13;
        hence thesis by A20,XBOOLE_0:def 5,A16;
        reconsider x1 = x as Element of l by A16;
      end;
      Fp c= tau Fp & ex q st q = A & q in tau F & q is conditional
      by LTLAXIO3: 16, A11;
      hence thesis by A14,XBOOLE_0:def 10;
    end;
  end;
