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 Th60:
  t |= f 'U' g iff ex m being Nat st (for j being Nat st j<m holds
  Shift(t,j) |= f) & Shift(t,m) |= g
proof
  set S1 = Inf_seq(S);
A1: f 'U' g = Until_0(f,g,S) by Def53;
A2: (ex m being Nat st (for j being Nat st j<m holds Shift(t,j) |= f) & (
  Shift(t,m) |= g )) implies t |= f 'U' g
  proof
    assume
A3: ex m being Nat st (for j being Nat st j<m holds Shift(t,j) |= f) &
    Shift(t,m) |= g;
    ex m being Nat st (for j being Nat st j<m holds Fid(f,S1).Shift(t,j,S
    )=TRUE) & Fid(g,S1).Shift(t,m,S) = TRUE
    proof
      consider m being Nat such that
A4:   for j being Nat st j<m holds Shift(t,j) |= f and
A5:   Shift(t,m) |= g by A3;
      take m;
      for j being Nat st j<m holds Fid(f,S1).Shift(t,j,S)=TRUE
      proof
        let j be Nat;
        assume j<m;
        then Shift(t,j) |= f by A4;
        hence thesis;
      end;
      hence thesis by A5;
    end;
    then Until_univ(t,Fid(f,S1),Fid(g,S1),S)=TRUE by Def51;
    then (Fid(f 'U' g,S1)).t=TRUE by A1,Def52;
    hence thesis;
  end;
  t |= f 'U' g implies ex m being Nat st (for j being Nat st j<m holds
  Shift(t,j) |= f) & Shift(t,m) |= g
  proof
    assume t|= f 'U' g;
    then (Fid(Until_0(f,g,S),S1)).t=TRUE by A1;
    then Until_univ(t,Fid(f,S1),Fid(g,S1),S)=TRUE by Def52;
    then consider m being Nat such that
A6: for j being Nat st j<m holds Fid(f,S1).Shift(t,j,S) =TRUE and
A7: Fid(g,S1).Shift(t,m,S) = TRUE by Def51;
    take m;
    for j being Nat st j<m holds Shift(t,j) |= f
    by A6;
    hence thesis by A7;
  end;
  hence thesis by A2;
end;
