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;
