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;
reserve CX for Consistent Subset of CQC-WFF(Al),
  P9 for Element of QC-pred_symbols(Al);
reserve JH for Henkin_interpretation of CX;

theorem Th15:
  |- f^<*VERUM(Al)*>
proof
  set PR = <*[f^<*VERUM(Al)*>,1]*>;
A1: rng PR = {[f^<*VERUM(Al)*>,1]} by FINSEQ_1:38;
  now
    let a be object;
    assume a in rng PR;
    then
A2: a = [f^<*VERUM(Al)*>,1] by A1,TARSKI:def 1;
    f^<*VERUM(Al)*> in set_of_CQC-WFF-seq(Al) by CALCUL_1:def 6;
    hence a in [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:] by A2,CQC_THE1:21
,ZFMISC_1:87;
  end;
  then rng PR c= [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
  then reconsider
  PR as FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds :] by
FINSEQ_1:def 4;
  now
    let n be Nat such that
A3: 1 <= n and
A4: n <= len PR;
    n <= 1 by A4,FINSEQ_1:39;
    then n = 1 by A3,XXREAL_0:1;
    then PR.n = [f^<*VERUM(Al)*>,1];
    then (PR.n)`1 = f^<*VERUM(Al)*> & (PR.n)`2 = 1;
    hence PR,n is_a_correct_step by CALCUL_1:def 7;
  end;
  then
A5: PR is a_proof by CALCUL_1:def 8;
  PR.1 = [f^<*VERUM(Al)*>,1];
  then PR.(len PR) = [f^<*VERUM(Al)*>,1] by FINSEQ_1:40;
  then (PR.(len PR))`1 = f^<*VERUM(Al)*>;
  hence thesis by A5,CALCUL_1:def 9;
end;
