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 Th44:
  for f,g,h1,h2 being Assign of BASSModel(R,BASSIGN) holds (for s
being Element of S holds s|= h1 implies s|= h2) implies for s being Element of
  S holds s|= Foax(g,f,h1) implies s|= Foax(g,f,h2)
proof
  let f,g,h1,h2 be Assign of BASSModel(R,BASSIGN);
  assume
A1: for s being Element of S holds s|= h1 implies s|= h2;
  let s be Element of S;
  assume
A2: s|= Foax(g,f,h1);
  per cases by A2,Th17;
  suppose
    s|= g;
    hence thesis by Th17;
  end;
  suppose
A3: s|= Fax(f,h1);
    then s|= EX(h1) by Th13;
    then consider pai be inf_path of R such that
A4: pai.0 = s and
A5: (pai.1) |= h1 by Th14;
    (pai.1) |= h2 by A1,A5;
    then
A6: s|= EX(h2) by A4,Th14;
    s|= f by A3,Th13;
    then s|= f '&' EX(h2) by A6,Th13;
    hence thesis by Th17;
  end;
end;
