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;

theorem Th32:
  N2 is_succ_of N1,F implies F in the LTLold of N2
proof
  assume
A1: N2 is_succ_of N1,F;
  now
    per cases by A1;
    suppose
      F in the LTLnew of N1 & N2 = SuccNode1(F,N1);
      then the LTLold of N2 =(the LTLold of N1) \/ {F} by Def4;
      then
A2:   {F} c= the LTLold of N2 by XBOOLE_1:7;
      F in {F} by TARSKI:def 1;
      hence thesis by A2;
    end;
    suppose
      F in the LTLnew of N1 & (F is disjunctive or F is Until or F is
      Release) & N2=SuccNode2(F,N1);
      then the LTLold of N2 =(the LTLold of N1) \/ {F} by Def5;
      then
A3:   {F} c= the LTLold of N2 by XBOOLE_1:7;
      F in {F} by TARSKI:def 1;
      hence thesis by A3;
    end;
  end;
  hence thesis;
end;
