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 Th39:
  s2 is_next_of s1 & H is Release & H in the LTLold of s2 & not
  the_left_argument_of H in the LTLold of s2 implies the_right_argument_of H in
  the LTLold of s2 & H in the LTLnext 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 LTLold of s2 and
A4: not F in the LTLold of s2;
  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;
  set m1 = m+1;
  set M2 = CastNode(L.m1,v);
  set n = len(L);
A9: CastNode(L.n,v) = s2 by A6,Def16;
  set M1 = CastNode(L.m,v);
A10: H in the LTLnew of M1 by A8;
A11: 1<=m1 & m1<=n by A7,NAT_1:13;
  then
A12: the LTLnext of M2 c= the LTLnext of s2 by A5,A9,Th31;
  the LTLnew of s2 = {} v by Def11;
  then
A13: the LTLnew of M2 c= the LTLold of s2 by A5,A9,A11,Th34;
  LTLNew2 H = {F,G} by A2,Def2;
  then
A14: F in LTLNew2 H by TARSKI:def 2;
A15: now
    the LTLold of M1 c= the LTLold of s2 by A5,A7,A9,Th31;
    then not F in the LTLold of M1 by A4;
    then F in LTLNew2 H \ the LTLold of M1 by A14,XBOOLE_0:def 5;
    then
A16: F in ((the LTLnew of M1) \ {H}) \/ (LTLNew2 H \ the LTLold of M1) by
XBOOLE_0:def 3;
    assume
A17: M2=SuccNode2(H,M1);
    not F in the LTLnew of M2 by A4,A13;
    hence contradiction by A10,A17,A16,Def5;
  end;
  LTLNew1 H = {G} by A2,Def1;
  then
A18: G in LTLNew1 H by TARSKI:def 1;
A19: M2 = SuccNode1(H,M1) or (H is disjunctive or H is Until or H is Release
  ) & M2=SuccNode2(H,M1) by A8;
A20: the LTLold of M2 c= the LTLold of s2 by A5,A9,A11,Th31;
A21: G in the LTLold of s2
  proof
    now
      per cases;
      suppose
        not G in the LTLold of M1;
        then G in LTLNew1 H \ the LTLold of M1 by A18,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 A10,A19,A15,Def4;
        hence thesis by A13;
      end;
      suppose
        G in the LTLold of M1;
        then G in (the LTLold of M1) \/ {H} by XBOOLE_0:def 3;
        then G in the LTLold of M2 by A10,A19,A15,Def4;
        hence thesis by A20;
      end;
    end;
    hence thesis;
  end;
  LTLNext H = {H} by A2,Def3;
  then H in LTLNext H by TARSKI:def 1;
  then H in (the LTLnext of M1) \/ LTLNext H by XBOOLE_0:def 3;
  then H in the LTLnext of M2 by A10,A19,A15,Def4;
  hence thesis by A12,A21;
end;
