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
  for f,g being Assign of BASSModel(R,BASSIGN), X being Subset of S holds
  (TransEU(f,g)).X = SIGMA(g) \/ (SIGMA(f) /\ Pred(X,R))
proof
  let f,g be Assign of BASSModel(R,BASSIGN);
  let X be Subset of S;
  set h = Tau(X,R,BASSIGN);
  (TransEU(f,g)).X = SigFoaxTau(g,f,X,R,BASSIGN) by Def73
    .= SIGMA(g) \/ SIGMA(Fax(f,h)) by Th33
    .= SIGMA(g) \/ (SIGMA(f) /\ SIGMA(EX(h))) by Th33
    .= SIGMA(g) \/ (SIGMA(f) /\ Pred(SIGMA(h),R)) by Th50;
  hence thesis by Th32;
end;
