theorem Th12:
  for f being Assign of BASSModel(R,BASSIGN) holds s |= 'not'(f) iff s |/= f
proof
  let f be Assign of BASSModel(R,BASSIGN);
A1: 'not' f = Not_0(f,S) by Def43;
A2: s |/= f implies s |= ('not' f)
  proof
    assume s |/= f;
    then not (Fid(f,S)).s =TRUE;
    then not Castboolean (Fid(f,S)).s = TRUE by Def4;
    then Castboolean (Fid(f,S)).s = FALSE by XBOOLEAN:def 3;
    then 'not' (Castboolean (Fid(f,S)).s) =TRUE;
    then (Fid('not'(f),S)).s =TRUE by A1,Def42;
    hence thesis;
  end;
  s |= 'not'(f) implies s |/= f
  proof
    assume s |= ('not' f);
    then (Fid(Not_0(f,S),S)).s = TRUE by A1;
    then 'not' (Castboolean (Fid(f,S)).s) =TRUE by Def42;
    then (Fid(f,S)).s = FALSE by Def4;
    hence thesis;
  end;
  hence thesis by A2;
end;
