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;
reserve BASSIGN for non empty Subset of ModelSP(Inf_seq(S));
reserve t for Element of Inf_seq(S);
reserve f,g for Assign of Inf_seqModel(S,BASSIGN);

theorem Th57:
  t |= 'not' f iff t |/= f
proof
  set S1 = Inf_seq(S);
A1: 'not' f = Not_0(f,S) by Def45;
  thus t |= 'not'(f) implies t |/= f
  proof
    assume t |= ('not' f);
    then (Fid('not' f,S1)).t=TRUE;
    then 'not' (Castboolean (Fid(f,S1)).t) = TRUE by A1,Def44;
    then (Fid(f,S1)).t = FALSE by MODELC_1:def 4;
    hence thesis;
  end;
  assume t |/= f;
  then not (Fid(f,S1)).t =TRUE;
  then not Castboolean (Fid(f,S1)).t = TRUE by MODELC_1:def 4;
  then Castboolean (Fid(f,S1)).t = FALSE by XBOOLEAN:def 3;
  then 'not' (Castboolean (Fid(f,S1)).t) =TRUE;
  then (Fid('not'(f),S1)).t =TRUE by A1,Def44;
  hence thesis;
end;
