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 Th72:
  r|= H1 'R' H2 iff r |= 'not' ( ('not' H1) 'U' ('not' H2))
proof
  set H01= Evaluate(H1,AtomicKai);
  set H02= Evaluate(H2,AtomicKai);
  set nH1= 'not' H1;
  set nH2= 'not' H2;
  r|= H1 'R' H2 iff r|= H01 'R' H02 by Th55;
  then
A1: r|= H1 'R' H2 iff r |= 'not' ( ('not' H01) 'U' ('not' H02)) by Def55;
  'not' H01 = Evaluate(nH1,AtomicKai) & 'not' H02 = Evaluate(nH2,AtomicKai
  ) by Th50;
  then r |= nH1 'U' nH2 iff r |= 'not' H01 'U' 'not' H02 by Th54;
  hence thesis by A1,Th57,Th64;
end;
