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;
reserve A for non empty set,
  v for Element of Valuations_in(Al,A),
  J for interpretation of Al,A;

theorem Th12:
  X is Inconsistent implies for J,v holds not J,v |= X
proof
  reconsider p = 'not' VERUM(Al) as Element of CQC-WFF(Al);
  assume not X is Consistent;
  then X |- 'not' VERUM(Al) by Th6;
  then consider f such that
A1: rng f c= X and
A2: |- f^<*'not' VERUM(Al)*>;
  let J,v;
A3: Suc(f^<*p*>) = p by CALCUL_1:5;
  rng Ant(f^<*p*>) c= X by A1,CALCUL_1:5;
  then p is_formal_provable_from X by A2,A3,CALCUL_1:def 10;
  then
A4: X |= p by CALCUL_1:32;
  now
    assume J,v |= X;
    then J,v |= 'not' VERUM(Al) by A4,CALCUL_1:def 12;
    then not J,v |= VERUM(Al) by VALUAT_1:17;
    hence contradiction by VALUAT_1:32;
  end;
  hence thesis;
end;
