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 Th62:
  t |= f 'R' g iff for m being Nat holds ((for j being Nat st j<m
  holds Shift(t,j) |= 'not'(f)) implies Shift(t,m) |= g)
proof
A1: (for m being Nat holds ((for j being Nat st j<m holds Shift(t,j) |=
'not'(f)) implies Shift(t,m) |/= 'not'(g))) implies for m being Nat holds ((for
  j being Nat st j<m holds Shift(t,j) |= 'not'(f)) implies Shift(t,m) |= g)
  proof
    assume
A2: for m being Nat holds ((for j being Nat st j<m holds Shift(t,j) |=
    'not'(f)) implies Shift(t,m) |/= 'not'(g));
    for m being Nat holds ((for j being Nat st j<m holds Shift(t,j) |=
    'not'(f)) implies Shift(t,m) |= g)
    proof
      let m be Nat;
      (for j being Nat st j<m holds Shift(t,j) |= 'not'(f)) implies Shift(
      t,m) |/= 'not'(g) by A2;
      hence thesis by Th57;
    end;
    hence thesis;
  end;
A3: (for m being Nat holds ((for j being Nat st j<m holds Shift(t,j) |= 'not'
(f)) implies Shift(t,m) |= g)) implies for m being Nat holds ((for j being Nat
  st j<m holds Shift(t,j) |= 'not'(f)) implies Shift(t,m) |/= 'not'(g))
  by Th57;
  t |= f 'R' g iff t|= 'not' ('not'(f) 'U' 'not'(g)) by Def55;
  then t |= f 'R' g iff not t|= 'not'(f) 'U' 'not'(g) by Th57;
  hence thesis by A1,A3,Th60;
end;
