reserve k,n for Element of NAT,
  a,Y for set,
  D,D1,D2 for non empty set,
  p,q for FinSequence of NAT;
reserve F,F1,G,G1,H,H1,H2 for CTL-formula;
reserve sq,sq9 for FinSequence;
reserve V for CTLModel;
reserve Kai for Function of atomic_WFF,the BasicAssign of V;
reserve f,f1,f2 for Function of CTL_WFF,the carrier of V;
reserve S for non empty set;
reserve R for total Relation of S,S;
reserve s,s0,s1 for Element of S;
reserve BASSIGN for non empty Subset of ModelSP(S);
reserve kai for Function of atomic_WFF,the BasicAssign of BASSModel(R,BASSIGN);

theorem Th16:
  for f,g being Assign of BASSModel(R,BASSIGN) holds s |= f EU g
iff ex pai being inf_path of R st pai.0 = s & ex m being Element of NAT st (for
  j being Element of NAT st j<m holds pai.j |= f) & pai.m |= g
proof
  let f,g be Assign of BASSModel(R,BASSIGN);
A1: f EU g = EUntill_0(f,g,R) by Def54;
A2: (ex pai being inf_path of R st pai.0 = s & ex m being Element of NAT st
(for j being Element of NAT st j<m holds (pai.j) |= f) & (pai.m) |= g) implies
  s |= f EU g
  proof
    assume
A3: ex pai being inf_path of R st pai.0 = s & ex m being Element of
NAT st (for j being Element of NAT st j<m holds (pai.j) |= f) & (pai.m) |= g;
    ex pai being inf_path of R st pai.0 = s & ex m being Element of NAT
st (for j being Element of NAT st j<m holds (Fid(f,S)).(pai.j) =TRUE) & (Fid(g,
    S)).(pai.m) =TRUE
    proof
      consider pai being inf_path of R such that
A4:   pai.0 = s and
A5:   ex m being Element of NAT st (for j being Element of NAT st j<m
      holds (pai.j) |= f) & (pai.m) |= g by A3;
      take pai;
      ex m being Element of NAT st (for j being Element of NAT st j<m
      holds (Fid(f,S)).(pai.j) =TRUE) & (Fid(g,S)).(pai.m) =TRUE
      proof
        consider m being Element of NAT such that
A6:     for j being Element of NAT st j<m holds (pai.j) |= f and
A7:     (pai.m) |= g by A5;
        take m;
        for j being Element of NAT st j<m holds (Fid(f,S)).(pai.j) =TRUE
        by A6,Def59;
        hence thesis by A7;
      end;
      hence thesis by A4;
    end;
    then EUntill_univ(s,Fid(f,S),Fid(g,S),R)=TRUE by Def52;
    then (Fid(f EU g,S)).s=TRUE by A1,Def53;
    hence thesis;
  end;
  s |= f EU g implies ex pai being inf_path of R st pai.0 = s & ex m being
Element of NAT st (for j being Element of NAT st j<m holds pai.j |= f) & pai.m
  |= g
  proof
    assume s|= f EU g;
    then (Fid(EUntill_0(f,g,R),S)).s=TRUE by A1;
    then EUntill_univ(s,Fid(f,S),Fid(g,S),R)=TRUE by Def53;
    then consider pai being inf_path of R such that
A8: pai.0 = s and
A9: ex m being Element of NAT st (for j being Element of NAT st j<m
    holds (Fid(f,S)).(pai.j) =TRUE ) & (Fid(g,S)).(pai.m) =TRUE by Def52;
    take pai;
    ex m being Element of NAT st (for j being Element of NAT st j<m holds
    (pai.j) |= f ) & (pai.m) |= g
    proof
      consider m being Element of NAT such that
A10:  for j being Element of NAT st j<m holds (Fid(f,S)).(pai.j) =TRUE and
A11:  (Fid(g,S)).(pai.m) =TRUE by A9;
      take m;
      for j being Element of NAT st j<m holds (pai.j) |= f
      by A10;
      hence thesis by A11;
    end;
    hence thesis by A8;
  end;
  hence thesis by A2;
end;
