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 Th14:
  for f being Assign of BASSModel(R,BASSIGN) holds s |= EX(f) iff
  ex pai being inf_path of R st pai.0 = s & (pai.1) |= f
proof
  let f be Assign of BASSModel(R,BASSIGN);
A1: EX(f) = EneXt_0(f,R) by Def46;
A2: (ex pai being inf_path of R st pai.0 = s & (pai.1) |= f) implies s |= EX
  (f)
  proof
    assume
A3: ex pai being inf_path of R st pai.0 = s & (pai.1) |= f;
    ex pai being inf_path of R st pai.0 = s & (Fid(f,S)).(pai.1) =TRUE
    by A3;
    then EneXt_univ(s,Fid(f,S),R)=TRUE by Def44;
    then (Fid(EX(f),S)).s=TRUE by A1,Def45;
    hence thesis;
  end;
  s |= EX(f) implies ex pai being inf_path of R st pai.0 = s & (pai.1) |= f
  proof
    assume s|= EX(f);
    then (Fid(EneXt_0(f,R),S)).s=TRUE by A1;
    then EneXt_univ(s,Fid(f,S),R)=TRUE by Def45;
    then consider pai being inf_path of R such that
A4: pai.0 = s and
A5: (Fid(f,S)).(pai.1) =TRUE by Def44;
    take pai;
    thus thesis by A4,A5;
  end;
  hence thesis by A2;
end;
