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 Th41:
  for f,g being Assign of BASSModel(R,BASSIGN) holds (for s being
  Element of S holds s|= g iff s|= Fax(f,g)) implies for s being Element of S
  holds s|= g implies s|= EG(f)
proof
  let f,g be Assign of BASSModel(R,BASSIGN);
  assume
A1: for s being Element of S holds s|= g iff s|= Fax(f,g);
A2: for s being Element of S holds s|= g implies s|= EX g
  proof
    let s be Element of S;
    assume s|= g;
    then s|= f '&' EX(g) by A1;
    hence thesis by Th13;
  end;
  for s0 being Element of S holds s0|= g implies s0|= EG(f)
  proof
    let s0 be Element of S;
    assume s0|= g;
    then consider pai be inf_path of R such that
A3: pai.0= s0 and
A4: for n being Nat holds pai.In(n,NAT) |= g by A2,Th40;
    for n being Element of NAT holds pai.n |= f
    proof
      let n be Element of NAT;
      pai.In(n,NAT) |= g by A4;
      then pai.n |= f '&' EX(g) by A1;
      hence thesis by Th13;
    end;
    hence thesis by A3,Th15;
  end;
  hence thesis;
end;
