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 Th58:
  t |= f '&' g iff t|=f & t|=g
proof
  set S1 = Inf_seq(S);
A1: f '&' g = And_0(f,g,S) by Def50;
  thus t |= f '&' g implies t|= f & t|= g
  proof
    assume t|= f '&' g;
    then (Fid(And_0(f,g,S),S1)).t=TRUE by A1;
    then
A2: (Castboolean (Fid(f,S1)).t) '&' (Castboolean (Fid(g,S1)).t) =TRUE by Def49;
    then Castboolean (Fid(g,S1)).t =TRUE by XBOOLEAN:101;
    then
A3: (Fid(g,S1)).t=TRUE by MODELC_1:def 4;
    Castboolean (Fid(f,S1)).t =TRUE by A2,XBOOLEAN:101;
    then (Fid(f,S1)).t =TRUE by MODELC_1:def 4;
    hence thesis by A3;
  end;
  assume t|= f & t|= g;
  then (Fid(f,S1)).t=TRUE & (Fid(g,S1)).t=TRUE;
  then (Castboolean (Fid(f,S1)).t) '&' (Castboolean (Fid(g,S1)).t) =TRUE by
MODELC_1:def 4;
  then (Fid(f '&' g,S1)).t=TRUE by A1,Def49;
  hence thesis;
end;
