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
  s,kai|= EG H iff ex pai being inf_path of R st pai.0 = s & for n being
  Element of NAT holds (pai.n),kai|= H
proof
A1: (ex pai being inf_path of R st pai.0 = s & for n being Element of NAT
holds (pai.n)|= Evaluate(H,kai) ) implies ex pai being inf_path of R st pai.0 =
  s & for n being Element of NAT holds (pai.n),kai|= H
  proof
    given pai be inf_path of R such that
A2: pai.0 = s and
A3: for n being Element of NAT holds (pai.n)|= Evaluate(H,kai);
    take pai;
    for n being Element of NAT holds (pai.n),kai|= H
    by A3;
    hence thesis by A2;
  end;
A4: (ex pai being inf_path of R st pai.0 = s & for n being Element of NAT
holds (pai.n),kai|= H ) implies ex pai being inf_path of R st pai.0 = s & for n
  being Element of NAT holds (pai.n)|= Evaluate(H,kai)
  proof
    given pai be inf_path of R such that
A5: pai.0 = s and
A6: for n being Element of NAT holds (pai.n),kai|= H;
    take pai;
    for n being Element of NAT holds (pai.n)|= Evaluate(H,kai)
    by A6,Def60;
    hence thesis by A5;
  end;
  s,kai|= EG H iff s|= EG Evaluate(H,kai) by Th8;
  hence thesis by A1,A4,Th15;
end;
