reserve k,n,m for Nat,
  a,x,X,Y for set,
  D,D1,D2,S for non empty set,
  p,q for FinSequence of NAT;
reserve F,F1,G,G1,H,H1,H2 for LTL-formula;
reserve sq,sq9 for FinSequence;
reserve L,L9 for FinSequence;
reserve j for Nat;
reserve j1 for Element of NAT;
reserve V for LTLModel;
reserve Kai for Function of atomic_LTL,the BasicAssign of V;
reserve f,f1,f2 for Function of LTL_WFF,the carrier of V;

theorem Th51:
  Evaluate(H1 '&' H2,Kai) = Evaluate(H1,Kai) '&' Evaluate(H2,Kai)
proof
  consider f0 be Function of LTL_WFF,the carrier of V such that
A1: f0 is-Evaluation-for Kai and
A2: Evaluate(H1 '&' H2,Kai) = f0.(H1 '&' H2) by Def35;
  consider f1 be Function of LTL_WFF,the carrier of V such that
A3: f1 is-Evaluation-for Kai and
A4: Evaluate(H1,Kai) = f1.H1 by Def35;
  consider f2 be Function of LTL_WFF,the carrier of V such that
A5: f2 is-Evaluation-for Kai and
A6: Evaluate(H2,Kai) = f2.H2 by Def35;
A7: f0=f2 by A1,A5,Th49;
A8: H1 '&' H2 is conjunctive;
  then
A9: the_left_argument_of(H1 '&' H2) = H1 & the_right_argument_of(H1 '&' H2)
  = H2 by Def19,Def20;
  f0=f1 by A1,A3,Th49;
  hence thesis by A1,A2,A4,A6,A7,A8,A9;
end;
