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;

theorem Th9:
  Evaluate(H1 EU H2,Kai) = Evaluate(H1,Kai) EU Evaluate(H2,Kai)
proof
  consider f0 be Function of CTL_WFF,the carrier of V such that
A1: f0 is-Evaluation-for Kai and
A2: Evaluate(H1 EU H2,Kai) = f0.(H1 EU H2) by Def34;
  consider f1 be Function of CTL_WFF,the carrier of V such that
A3: f1 is-Evaluation-for Kai and
A4: Evaluate(H1,Kai) = f1.H1 by Def34;
  consider f2 be Function of CTL_WFF,the carrier of V such that
A5: f2 is-Evaluation-for Kai and
A6: Evaluate(H2,Kai) = f2.H2 by Def34;
A7: f0=f2 by A1,A5,Th4;
A8: H1 EU H2 is ExistUntill;
  then (H1 EU H2).1 = 4 by Lm7;
  then
A9: not H1 EU H2 is conjunctive by Lm4;
  then
A10: the_left_argument_of(H1 EU H2) = H1 by A8,Def22;
A11: the_right_argument_of(H1 EU H2) = H2 by A8,A9,Def23;
  f0=f1 by A1,A3,Th4;
  hence thesis by A1,A2,A4,A6,A7,A8,A10,A11;
end;
