reserve k,n,n1,m,m1,m0,h,i,j for Nat,
  a,x,y,X,X1,X2,X3,X4,Y for set;
reserve L,L1,L2 for FinSequence;
reserve F,F1,G,G1,H for LTL-formula;
reserve W,W1,W2 for Subset of Subformulae H;
reserve v for LTL-formula;
reserve N,N1,N2,N10,N20,M for strict LTLnode over v;
reserve w for Element of Inf_seq(AtomicFamily);
reserve R1,R2 for Real_Sequence;
reserve s,s0,s1,s2 for elementary strict LTLnode over v;

theorem Th40:
  s2 is_next_of s1 & H is Release & H in the LTLnext of s1 implies
  the_right_argument_of H in the LTLold of s2 & H in the LTLold of s2
proof
  set F = the_left_argument_of H;
  set G = the_right_argument_of H;
  set N1 = 'X' s1;
  assume that
A1: s2 is_next_of s1 and
A2: H is Release and
A3: H in the LTLnext of s1;
A4: the LTLnext of s1 c= the LTLold of s2 by A1,Th37;
  then consider L,m such that
  1<=len(L) and
A5: L is_Finseq_for v and
  L.1 = N1 and
A6: L.(len(L)) = s2 and
A7: 1<= m & m <len(L) and
A8: CastNode(L.(m+1),v) is_succ_of CastNode(L.m,v),H by A1,A3,Th38;
A9: the LTLnew of s2 = {} v by Def11;
  set M1 = CastNode(L.m,v);
  set m1 = m+1;
  set M2 = CastNode(L.m1,v);
  set n = len(L);
A10: CastNode(L.n,v) = s2 by A6,Def16;
  1<=m1 & m1<=n by A7,NAT_1:13;
  then
A11: the LTLnew of M2 c= the LTLold of s2 by A5,A10,A9,Th34;
  LTLNew2 H = {F,G} by A2,Def2;
  then
A12: G in LTLNew2 H by TARSKI:def 2;
  LTLNew1 H = {G} by A2,Def1;
  then
A13: G in LTLNew1 H by TARSKI:def 1;
A14: the LTLold of M1 c= the LTLold of s2 by A5,A7,A10,Th31;
  G in the LTLold of s2
  proof
    now
      per cases;
      suppose
        G in the LTLold of M1;
        hence thesis by A14;
      end;
      suppose
A15:    not G in the LTLold of M1;
        now
          per cases by A8;
          suppose
A16:        H in the LTLnew of M1 & M2 = SuccNode1(H,M1);
            G in LTLNew1 H \ the LTLold of M1 by A13,A15,XBOOLE_0:def 5;
            then
            G in ((the LTLnew of M1) \ {H}) \/ (LTLNew1 H \ the LTLold of
            M1) by XBOOLE_0:def 3;
            then G in the LTLnew of M2 by A16,Def4;
            hence thesis by A11;
          end;
          suppose
A17:        H in the LTLnew of M1 & (H is disjunctive or H is Until or
            H is Release) & M2=SuccNode2(H,M1);
            G in LTLNew2 H \ the LTLold of M1 by A12,A15,XBOOLE_0:def 5;
            then
            G in ((the LTLnew of M1) \ {H}) \/ (LTLNew2 H \ the LTLold of
            M1) by XBOOLE_0:def 3;
            then G in the LTLnew of M2 by A17,Def5;
            hence thesis by A11;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  hence thesis by A3,A4;
end;
