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 Th13:
  for f,g being Assign of BASSModel(R,BASSIGN) holds s |= f '&' g
  iff s|=f & s|=g
proof
  let f,g be Assign of BASSModel(R,BASSIGN);
A1: f '&' g = And_0(f,g,S) by Def51;
A2: s |= f '&' g implies s|= f & s|= g
  proof
    assume s|= f '&' g;
    then (Fid(And_0(f,g,S),S)).s=TRUE by A1;
    then
A3: (Castboolean (Fid(f,S)).s) '&' (Castboolean (Fid(g,S)).s) =TRUE by Def50;
    then Castboolean (Fid(g,S)).s=TRUE by XBOOLEAN:101;
    then
A4: (Fid(g,S)).s=TRUE by Def4;
    Castboolean (Fid(f,S)).s =TRUE by A3,XBOOLEAN:101;
    then (Fid(f,S)).s =TRUE by Def4;
    hence thesis by A4;
  end;
  s|= f & s|= g implies s |= f '&' g
  proof
    assume that
A5: s|= f and
A6: s|= g;
A7: (Fid(g,S)).s=TRUE by A6;
    (Fid(f,S)).s=TRUE by A5;
    then (Castboolean (Fid(f,S)).s) '&' (Castboolean (Fid(g,S)).s) =TRUE by A7
,Def4;
    then (Fid(f '&' g,S)).s=TRUE by A1,Def50;
    hence thesis;
  end;
  hence thesis by A2;
end;
