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);

theorem Th59:
  t |= 'X' f iff Shift(t,1) |=f
proof
  set S1 = Inf_seq(S);
  set t1 = Shift(t,1);
  set t1p = Shift(t,1,S);
A1: 'X' f = Next_0(f,S) by Def48;
  thus t |= 'X' f implies t1 |=f
  proof
    assume t|= 'X' f;
    then (Fid(Next_0(f,S),S1)).t=TRUE by A1;
    then Next_univ(t,Fid(f,S1))=TRUE by Def47;
    then Fid(f,S1).t1p =TRUE by Def46;
    hence thesis;
  end;
  assume t1 |=f;
  then Fid(f,S1).t1 = TRUE;
  then Next_univ(t,Fid(f,S1))=TRUE by Def46;
  then (Fid('X' f,S1)).t=TRUE by A1,Def47;
  hence thesis;
end;
