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 Th41:
  s1 is_next_of s0 & H in the LTLold of s1 implies (H is
  conjunctive implies the_left_argument_of H in the LTLold of s1 &
the_right_argument_of H in the LTLold of s1) & (H is disjunctive or H is Until
  implies the_left_argument_of H in the LTLold of s1 or the_right_argument_of H
  in the LTLold of s1) & (H is next implies the_argument_of H in the LTLnext of
  s1 ) & (H is Release implies the_right_argument_of H in the LTLold of s1)
proof
  assume that
A1: s1 is_next_of s0 and
A2: H in the LTLold of s1;
  consider L such that
A3: 1<=len(L) and
A4: L is_Finseq_for v and
A5: L.1 = 'X' s0 and
A6: L.(len(L)) = s1 by A1;
A7: CastNode(L.1,v) = 'X' s0 by A5,Def16;
  set n = len(L);
A8: CastNode(L.n,v) = s1 by A6,Def16;
  1<n by A2,A3,A5,A6,XXREAL_0:1;
  then consider m such that
A9: 1<= m & m<n and
A10: ( not H in the LTLold of CastNode(L.m,v))& H in the LTLold of
  CastNode(L.(m+1) ,v) by A2,A4,A8,A7,Th27;
  consider N1,N2 such that
A11: N1 = L.m and
A12: N2 = L.(m+1) and
A13: N2 is_succ_of N1 by A4,A9;
A14: CastNode(L.m,v) = N1 by A11,Def16;
  then
A15: the LTLold of N1 c= the LTLold of s1 by A4,A8,A9,Th31;
  set m1=m+1;
A16: m1<=n & 1<=m1 by A9,NAT_1:13;
A17: CastNode(L.(m+1),v) = N2 by A12,Def16;
  then
A18: N2 is_succ_of N1,H by A10,A13,A14,Th28;
  the LTLnew of CastNode(L.n,v) = {} v by A8,Def11;
  then
A19: the LTLnew of N2 c= the LTLold of s1 by A4,A8,A17,A16,Th34;
A20: H is conjunctive implies the_left_argument_of H in the LTLold of s1 &
  the_right_argument_of H in the LTLold of s1
  proof
    set G = the_right_argument_of H;
    set F = the_left_argument_of H;
    assume
A21: H is conjunctive;
    then
A22: LTLNew1 H = {F,G} by Def1;
    now
      per cases by A18;
      suppose
        H in the LTLnew of N1 & N2 = SuccNode1(H,N1);
        then
        the LTLnew of N2 = ((the LTLnew of N1) \ {H}) \/ (LTLNew1 H \ the
        LTLold of N1) by Def4;
        then
A23:    (LTLNew1 H \ the LTLold of N1) c= the LTLnew of N2 by XBOOLE_1:7;
A24:    G in the LTLold of s1
        proof
          now
            per cases;
            suppose
              G in the LTLold of N1;
              hence thesis by A15;
            end;
            suppose
A25:          not G in the LTLold of N1;
              G in LTLNew1 H by A22,TARSKI:def 2;
              then G in LTLNew1 H \ the LTLold of N1 by A25,XBOOLE_0:def 5;
              then G in the LTLnew of N2 by A23;
              hence thesis by A19;
            end;
          end;
          hence thesis;
        end;
        F in the LTLold of s1
        proof
          now
            per cases;
            suppose
              F in the LTLold of N1;
              hence thesis by A15;
            end;
            suppose
A26:          not F in the LTLold of N1;
              F in LTLNew1 H by A22,TARSKI:def 2;
              then F in LTLNew1 H \ the LTLold of N1 by A26,XBOOLE_0:def 5;
              then F in the LTLnew of N2 by A23;
              hence thesis by A19;
            end;
          end;
          hence thesis;
        end;
        hence thesis by A24;
      end;
      suppose
        H in the LTLnew of N1 & (H is disjunctive or H is Until or H
        is Release) & N2=SuccNode2(H,N1);
        hence thesis by A21,MODELC_2:78;
      end;
    end;
    hence thesis;
  end;
A27: H is Release implies the_right_argument_of H in the LTLold of s1
  proof
    set G = the_right_argument_of H;
    set F = the_left_argument_of H;
    assume
A28: H is Release;
    then
A29: LTLNew2 H = {F,G} by Def2;
A30: LTLNew1 H = {G} by A28,Def1;
    now
      per cases by A18;
      suppose
        H in the LTLnew of N1 & N2 = SuccNode1(H,N1);
        then
        the LTLnew of N2 = ((the LTLnew of N1) \ {H}) \/ (LTLNew1 H \ the
        LTLold of N1) by Def4;
        then
A31:    (LTLNew1 H \ the LTLold of N1) c= the LTLnew of N2 by XBOOLE_1:7;
        G in the LTLold of s1
        proof
          now
            per cases;
            suppose
              G in the LTLold of N1;
              hence thesis by A15;
            end;
            suppose
A32:          not G in the LTLold of N1;
              G in LTLNew1 H by A30,TARSKI:def 1;
              then G in LTLNew1 H \ the LTLold of N1 by A32,XBOOLE_0:def 5;
              then G in the LTLnew of N2 by A31;
              hence thesis by A19;
            end;
          end;
          hence thesis;
        end;
        hence thesis;
      end;
      suppose
        H in the LTLnew of N1 & (H is disjunctive or H is Until or H
        is Release) & N2=SuccNode2(H,N1);
        then
        the LTLnew of N2 = ((the LTLnew of N1) \ {H}) \/ (LTLNew2 H \ the
        LTLold of N1) by Def5;
        then
A33:    (LTLNew2 H \ the LTLold of N1) c= the LTLnew of N2 by XBOOLE_1:7;
        G in the LTLold of s1
        proof
          now
            per cases;
            suppose
              G in the LTLold of N1;
              hence thesis by A15;
            end;
            suppose
A34:          not G in the LTLold of N1;
              G in LTLNew2 H by A29,TARSKI:def 2;
              then G in LTLNew2 H \ the LTLold of N1 by A34,XBOOLE_0:def 5;
              then G in the LTLnew of N2 by A33;
              hence thesis by A19;
            end;
          end;
          hence thesis;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
A35: H is disjunctive or H is Until implies the_left_argument_of H in the
  LTLold of s1 or the_right_argument_of H in the LTLold of s1
  proof
    set G = the_right_argument_of H;
    set F = the_left_argument_of H;
    assume
A36: H is disjunctive or H is Until;
    then
A37: LTLNew2 H = {G} by Def2;
A38: LTLNew1 H = {F} by A36,Def1;
    now
      per cases by A18;
      suppose
        H in the LTLnew of N1 & N2 = SuccNode1(H,N1);
        then
        the LTLnew of N2 = ((the LTLnew of N1) \ {H}) \/ (LTLNew1 H \ the
        LTLold of N1) by Def4;
        then
A39:    (LTLNew1 H \ the LTLold of N1) c= the LTLnew of N2 by XBOOLE_1:7;
        F in the LTLold of s1
        proof
          now
            per cases;
            suppose
              F in the LTLold of N1;
              hence thesis by A15;
            end;
            suppose
A40:          not F in the LTLold of N1;
              F in LTLNew1 H by A38,TARSKI:def 1;
              then F in LTLNew1 H \ the LTLold of N1 by A40,XBOOLE_0:def 5;
              then F in the LTLnew of N2 by A39;
              hence thesis by A19;
            end;
          end;
          hence thesis;
        end;
        hence thesis;
      end;
      suppose
        H in the LTLnew of N1 & (H is disjunctive or H is Until or H
        is Release) & N2=SuccNode2(H,N1);
        then
        the LTLnew of N2 = ((the LTLnew of N1) \ {H}) \/ (LTLNew2 H \ the
        LTLold of N1) by Def5;
        then
A41:    (LTLNew2 H \ the LTLold of N1) c= the LTLnew of N2 by XBOOLE_1:7;
        G in the LTLold of s1
        proof
          now
            per cases;
            suppose
              G in the LTLold of N1;
              hence thesis by A15;
            end;
            suppose
A42:          not G in the LTLold of N1;
              G in LTLNew2 H by A37,TARSKI:def 1;
              then G in LTLNew2 H \ the LTLold of N1 by A42,XBOOLE_0:def 5;
              then G in the LTLnew of N2 by A41;
              hence thesis by A19;
            end;
          end;
          hence thesis;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
A43: the LTLnext of N2 c= the LTLnext of s1 by A4,A8,A17,A16,Th31;
  H is next implies the_argument_of H in the LTLnext of s1
  proof
    set F = the_argument_of H;
    assume
A44: H is next;
    then
A45: LTLNext H = {F} by Def3;
    now
      per cases by A18;
      suppose
        H in the LTLnew of N1 & N2 = SuccNode1(H,N1);
        then the LTLnext of N2 = (the LTLnext of N1) \/ LTLNext H by Def4;
        then LTLNext H c= the LTLnext of N2 by XBOOLE_1:7;
        then
A46:    LTLNext H c= the LTLnext of s1 by A43;
        F in LTLNext H by A45,TARSKI:def 1;
        hence thesis by A46;
      end;
      suppose
        H in the LTLnew of N1 & (H is disjunctive or H is Until or H
        is Release) & N2=SuccNode2(H,N1);
        hence thesis by A44,MODELC_2:78;
      end;
    end;
    hence thesis;
  end;
  hence thesis by A20,A35,A27;
end;
