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 Th68:
  r|= H1 'U' H2 iff ex m being Nat st (for j being Nat st j<m
  holds Shift(r,j) |= H1) & Shift(r,m) |= H2
proof
A1: (ex m being Nat st (for j being Nat st j<m holds Shift(r,j) |= Evaluate(
  H1,AtomicKai)) & (Shift(r,m) |= Evaluate(H2,AtomicKai) )) implies ex m being
  Nat st (for j being Nat st j<m holds Shift(r,j) |= H1) & Shift(r,m) |= H2
  proof
    assume ex m being Nat st (for j being Nat st j<m holds Shift(r,j) |=
    Evaluate(H1,AtomicKai)) & Shift(r,m) |= Evaluate(H2,AtomicKai);
    then consider m be Nat such that
A2: for j being Nat st j<m holds Shift(r,j) |= Evaluate(H1,AtomicKai) and
A3: Shift(r,m) |= Evaluate(H2,AtomicKai);
    take m;
    for j being Nat st j<m holds Shift(r,j) |= H1
    by A2;
    hence thesis by A3;
  end;
A4: (ex m being Nat st (for j being Nat st j<m holds Shift(r,j) |= H1) & (
  Shift(r,m) |= H2)) implies ex m being Nat st (for j being Nat st j<m holds
  Shift(r,j) |= Evaluate(H1,AtomicKai)) & Shift(r,m) |= Evaluate(H2,AtomicKai)
  proof
    assume ex m being Nat st (for j being Nat st j<m holds Shift(r,j) |= H1)
    & Shift(r,m) |= H2;
    then consider m be Nat such that
A5: for j being Nat st j<m holds Shift(r,j) |= H1 and
A6: Shift(r,m) |= H2;
    take m;
    for j being Nat st j<m holds Shift(r,j) |= Evaluate(H1,AtomicKai)
    by A5,Def63;
    hence thesis by A6;
  end;
  r|= H1 'U' H2 iff r|= Evaluate(H1,AtomicKai) 'U' Evaluate(H2,AtomicKai)
  by Th54;
  hence thesis by A1,A4,Th60;
end;
