theorem Th4:
  X is Consistent & rng g c= X implies g is Consistent
proof
  assume that
A1: X is Consistent and
A2: rng g c= X;
  now
    assume g is Inconsistent;
    then consider p such that
A3: |- g^<*p*> & |- g^<*'not' p*>;
    X |- p & X |- 'not' p by A2,A3;
    hence contradiction by A1;
  end;
  hence thesis;
end;
