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|= H1 EU H2 iff ex pai being inf_path of R st pai.0 = s & ex m
being Element of NAT st (for j being Element of NAT st j<m holds pai.j,kai|= H1
  ) & pai.m,kai|= H2
proof
A1: (ex pai being inf_path of R st pai.0 = s & ex m being Element of NAT st
(for j being Element of NAT st j<m holds pai.j |= Evaluate(H1,kai)) & (pai.m |=
  Evaluate(H2,kai)) ) implies ex pai being inf_path of R st pai.0 = s & ex m
being Element of NAT st (for j being Element of NAT st j<m holds pai.j,kai|= H1
  ) & pai.m,kai|= H2
  proof
    given pai be inf_path of R such that
A2: pai.0 = s and
A3: ex m being Element of NAT st (for j being Element of NAT st j<m
    holds pai.j |= Evaluate(H1,kai)) & pai.m |= Evaluate(H2,kai);
    take pai;
    consider m be Element of NAT such that
A4: for j being Element of NAT st j<m holds pai.j |= Evaluate(H1,kai) and
A5: pai.m |= Evaluate(H2,kai) by A3;
A6: for j being Element of NAT st j<m holds pai.j,kai|= H1
    by A4;
    pai.m |= Evaluate(H2,kai) iff pai.m,kai|= H2;
    hence thesis by A2,A5,A6;
  end;
A7: (ex pai being inf_path of R st pai.0 = s & ex m being Element of NAT st
  (for j being Element of NAT st j<m holds pai.j,kai|= H1) & (pai.m,kai|= H2) )
implies ex pai being inf_path of R st pai.0 = s & ex m being Element of NAT st
(for j being Element of NAT st j<m holds pai.j |= Evaluate(H1,kai)) & pai.m |=
  Evaluate(H2,kai)
  proof
    given pai be inf_path of R such that
A8: pai.0 = s and
A9: ex m being Element of NAT st (for j being Element of NAT st j<m
    holds pai.j,kai|= H1) & pai.m,kai|= H2;
    take pai;
    consider m be Element of NAT such that
A10: for j being Element of NAT st j<m holds pai.j,kai|= H1 and
A11: pai.m,kai|= H2 by A9;
A12: for j being Element of NAT st j<m holds pai.j |= Evaluate(H1,kai)
    by A10,Def60;
    thus thesis by A8,A11,A12;
  end;
  s,kai|= H1 EU H2 iff s|= Evaluate(H1,kai) EU Evaluate(H2,kai) by Th9;
  hence thesis by A1,A7,Th16;
end;
