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 Th15:
  for f being Assign of BASSModel(R,BASSIGN) holds s |= EG(f) iff
ex pai being inf_path of R st pai.0 = s & for n being Element of NAT holds (pai
  .n) |= f
proof
  let f be Assign of BASSModel(R,BASSIGN);
A1: EG(f) = EGlobal_0(f,R) by Def49;
A2: (ex pai being inf_path of R st pai.0 = s & for n being Element of NAT
  holds (pai.n) |= f) implies s |= EG(f)
  proof
    assume
A3: ex pai being inf_path of R st pai.0 = s & for n being Element of
    NAT holds (pai.n) |= f;
    ex pai being inf_path of R st pai.0 = s & for n being Element of NAT
    holds (Fid(f,S)).(pai.n) =TRUE
    proof
      consider pai being inf_path of R such that
A4:   pai.0 = s and
A5:   for n being Element of NAT holds (pai.n) |= f by A3;
      take pai;
      for n being Element of NAT holds (Fid(f,S)).(pai.n) =TRUE
      by A5,Def59;
      hence thesis by A4;
    end;
    then EGlobal_univ(s,Fid(f,S),R)=TRUE by Def47;
    then (Fid(EG(f),S)).s=TRUE by A1,Def48;
    hence thesis;
  end;
  s |= EG(f) implies ex pai being inf_path of R st pai.0 = s & for n being
  Element of NAT holds (pai.n) |= f
  proof
    assume s|= EG(f);
    then (Fid(EGlobal_0(f,R),S)).s=TRUE by A1;
    then EGlobal_univ(s,Fid(f,S),R)=TRUE by Def48;
    then consider pai being inf_path of R such that
A6: pai.0 = s and
A7: for n being Element of NAT holds (Fid(f,S)).(pai.n) =TRUE by Def47;
    take pai;
    for n being Element of NAT holds (pai.n) |= f
    by A7;
    hence thesis by A6;
  end;
  hence thesis by A2;
end;
