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 Th6:
  X is Inconsistent iff for p holds X |- p
proof
  thus X is Inconsistent implies for p holds X |- p
  proof
    assume X is Inconsistent;
    then consider q such that
A1: X |- q and
A2: X |- 'not' q;
    consider f2 such that
A3: rng f2 c= X and
A4: |- f2^<*'not' q*> by A2;
    let p;
    consider f1 such that
A5: rng f1 c= X and
A6: |- f1^<*q*> by A1;
    take f3 = f1^f2;
A7: rng f3 = rng f1 \/ rng f2 by FINSEQ_1:31;
A8: |- f1^f2^<*'not' q*> by A4,CALCUL_2:20;
    |- f1^f2^<*q*> by A6,Th5;
    hence thesis by A5,A3,A8,A7,CALCUL_2:25,XBOOLE_1:8;
  end;
  assume for p holds X |- p;
  then X |- VERUM(Al) & X |- 'not' VERUM(Al);
  hence thesis;
end;
