reserve A for QC-alphabet;
reserve i,j,k for Nat;
reserve f for Substitution of A;
reserve x,y for bound_QC-variable of A;
reserve a for free_QC-variable of A;
reserve p,q for Element of QC-WFF(A);
reserve l,l1,l2,ll for FinSequence of QC-variables(A);
reserve r,s for Element of CQC-WFF(A);

theorem Th26:
  Free p = {} implies p.x = p
proof
  defpred P[Element of QC-WFF(A)] means Free $1 = {} implies $1.x = $1;
A1: for p st P[p] holds P['not' p] by Th19,QC_LANG3:55;
A2: for p,q st P[p] & P[q] holds P[p '&' q]
  proof
    let p,q;
    assume
A3: ( Free p = {} implies p.x = p)&( Free q = {} implies q.x = q);
    assume Free(p '&' q) = {};
    then (Free p) \/ (Free q) = {} by QC_LANG3:57;
    hence thesis by A3,Th21;
  end;
A4: for k being Nat
  for P being (QC-pred_symbol of k,A),l being QC-variable_list of k, A
  holds P[P!l]
  proof
    let k be Nat;
    let P be (QC-pred_symbol of k,A),l be QC-variable_list of k, A;
    assume
A5: Free(P!l) = {};
A6: now
      let j be Nat;
      assume
A7:   1 <= j & j <= len l;
      now
        assume l.j = (A)a.0;
        then (A)a.0 in { l.i where i is Nat : 1 <= i &
            i <= len l & l.i in free_QC-variables(A) }
        by A7;
        hence contradiction by A5,QC_LANG3:54;
      end;
      hence Subst(l,(A)a.0.-->x).j = l.j by A7,Th3;
    end;
    len Subst(l,(A)a.0.-->x) = len l by Def1;
    then Subst(l,(A)a.0.-->x) = l by A6,FINSEQ_1:14;
    hence thesis by Th17;
  end;
A8: for y,p st P[p] holds P[All(y, p)]
  proof
    let y,p;
    assume
A9: Free p = {} implies p.x = p;
A10: x = y implies All(y, p).x = All(y, p) by Th24;
    assume Free All(y,p) = {};
    hence thesis by A9,A10,Th25,QC_LANG3:58;
  end;
A11: P[VERUM(A)] by Th15;
  for p holds P[p] from QC_LANG1:sch 1(A4,A11,A1,A2,A8);
  hence thesis;
end;
