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 Th25:
  N2 is_succ_of N1 implies the LTLold of N1 c= the LTLold of N2 &
  the LTLnext of N1 c= the LTLnext of N2
proof
  assume
A1: N2 is_succ_of N1;
  now
    per cases by A1;
    suppose
      N2 is_succ1_of N1;
      then consider H such that
A2:   H in the LTLnew of N1 & N2 = SuccNode1(H,N1);
      the LTLold of N2 =(the LTLold of N1) \/ {H} & the LTLnext of N2 = (
      the LTLnext of N1) \/ LTLNext H by A2,Def4;
      hence thesis by XBOOLE_1:7;
    end;
    suppose
      N2 is_succ2_of N1;
      then consider H such that
A3:   H in the LTLnew of N1 and
      H is disjunctive or H is Until or H is Release and
A4:   N2 = SuccNode2(H,N1);
      the LTLold of N2 =(the LTLold of N1) \/ {H} by A3,A4,Def5;
      hence thesis by A3,A4,Def5,XBOOLE_1:7;
    end;
  end;
  hence thesis;
end;
