reserve Al for QC-alphabet;
reserve a,a1,a2,b,c,d for set,
  X,Y,Z for Subset of CQC-WFF(Al),
  i,k,m,n for Nat,
  p,q for Element of CQC-WFF(Al),
  P for QC-pred_symbol of k,Al,
  ll for CQC-variable_list of k,Al,
  f,f1,f2,g for FinSequence of CQC-WFF(Al);
reserve A for non empty finite Subset of NAT;
reserve C for non empty set;

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;
