reserve k,n,m for Nat,
  a,x,X,Y for set,
  D,D1,D2,S for non empty set,
  p,q for FinSequence of NAT;
reserve F,F1,G,G1,H,H1,H2 for LTL-formula;
reserve sq,sq9 for FinSequence;
reserve L,L9 for FinSequence;
reserve j for Nat;
reserve j1 for Element of NAT;
reserve V for LTLModel;
reserve Kai for Function of atomic_LTL,the BasicAssign of V;
reserve f,f1,f2 for Function of LTL_WFF,the carrier of V;
reserve BASSIGN for non empty Subset of ModelSP(Inf_seq(S));
reserve t for Element of Inf_seq(S);
reserve f,g for Assign of Inf_seqModel(S,BASSIGN);
reserve r for Element of Inf_seq(AtomicFamily);
reserve W for Subset of LTL_WFF;

theorem
  r |= 'X' W iff Shift(r,1) |=W
proof
A1: Shift(r,1) |=W implies r |= 'X' W
  proof
    assume
A2: Shift(r,1) |=W;
A3: for u be LTL-formula st u in W holds r|= 'X' u
    by A2,Th67;
    for H be LTL-formula st H in 'X' W holds r |= H
    proof
      let H being LTL-formula;
      assume H in 'X' W;
      then
      ex x being LTL-formula st H = x & ex u being LTL-formula st u in W &
      x='X' u;
      hence thesis by A3;
    end;
    hence thesis;
  end;
  r |= 'X' W implies Shift(r,1) |=W
  proof
    assume
A4: r |= 'X' W;
    for H be LTL-formula st H in W holds Shift(r,1) |= H
    proof
      let H being LTL-formula;
      set u = 'X' H;
      assume H in W;
      then u in 'X' W;
      then r |= u by A4;
      hence thesis by Th67;
    end;
    hence thesis;
  end;
  hence thesis by A1;
end;
