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);

theorem
  r|= H1 'R' H2 iff r|=(H1 '&' H2) 'or' (H2 '&' ('X' (H1 'R' H2)))
proof
  set nH1= 'not' H1;
  set nH2= 'not' H2;
  r|= H1 'R' H2 iff r |= 'not' ( nH1 'U' nH2) by Th72;
  then r|= H1 'R' H2 iff r |/= nH1 'U' nH2 by Th64;
  then r|= H1 'R' H2 iff r |/= nH2 'or' (nH1 '&' ('X' (nH1 'U' nH2))) by Th75;
  then r|= H1 'R' H2 iff r |= 'not'( nH2 'or' (nH1 '&' ('X' (nH1 'U' nH2))) )
  by Th64;
  then
  r|= H1 'R' H2 iff r |= 'not'(nH2) '&' ('not'(nH1 '&' ('X' (nH1 'U' nH2))
  )) by Th70;
  then
  r|= H1 'R' H2 iff r |= 'not'(nH2) & r |= 'not'(nH1 '&' ('X' (nH1 'U' nH2
  ))) by Th65;
  then r|= H1 'R' H2 iff r |/= nH2 & r |= 'not'(nH1) 'or' ('not'('X' (nH1 'U'
  nH2))) by Th64,Th71;
  then
  r|= H1 'R' H2 iff r |= H2 & ( r |= 'not'(nH1) or r |='not'('X' (nH1 'U'
  nH2))) by Th64,Th66;
  then
  r|= H1 'R' H2 iff r |= H2 & ( r |/= nH1 or r |='X' ('not'(nH1 'U' nH2)))
  by Th64,Th74;
  then r|= H1 'R' H2 iff r |= H2 & ( r |/= nH1 or Shift(r,1) |='not'(nH1 'U'
  nH2)) by Th67;
  then r|= H1 'R' H2 iff r |= H2 & ( r |= H1 or Shift(r,1)|=H1 'R' H2 ) by Th64
,Th72;
  then r|= H1 'R' H2 iff r |= H2 & ( r |= H1 or r |='X' (H1 'R' H2)) by Th67;
  then r|= H1 'R' H2 iff r |= H1 '&' H2 or r |= H2 '&' 'X' (H1 'R' H2) by Th65;
  hence thesis by Th66;
end;
