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
  Fixed(p.x) = Fixed p
proof
  defpred P[Element of QC-WFF(A)] means Fixed($1.x) = Fixed $1;
A1: for p st P[p] holds P['not' p]
  proof
    let p such that
A2: Fixed (p.x) = Fixed p;
    thus Fixed(('not' p).x) = Fixed('not' (p.x)) by Th19
      .= Fixed p by A2,QC_LANG3:65
      .= Fixed 'not' p by QC_LANG3:65;
  end;
A3: for p,q st P[p] & P[q] holds P[p '&' q]
  proof
    let p,q such that
A4: Fixed(p.x) = Fixed(p) & Fixed(q.x) = Fixed q;
    thus Fixed((p '&' q).x) = Fixed((p.x) '&' (q.x)) by Th21
      .= Fixed(p) \/ Fixed(q) by A4,QC_LANG3:67
      .= Fixed (p '&' q) by QC_LANG3:67;
  end;
A5: for k for P being (QC-pred_symbol of k,A),l being QC-variable_list of k, A
  holds P[P!l]
  proof
    let k;
    let P be (QC-pred_symbol of k,A),l be QC-variable_list of k, A;
    set F1 = { l.i : 1 <= i & i <= len l & l.i in fixed_QC-variables(A) };
    set ll = Subst(l,(A)a.0.-->x);
    set F2 = { ll.i : 1 <= i & i <= len ll & ll.i in fixed_QC-variables(A) };
A6: len l = len ll by Def1;
    now
      let y be object;
      thus y in F1 implies y in F2
      proof
        assume y in F1;
        then consider i such that
A7:     y = l.i and
A8:     1 <= i & i <= len l and
A9:     l.i in fixed_QC-variables(A);
        l.i <> (A)a.0 by A9,QC_LANG3:32;
        then ll.i = l.i by A8,Th3;
        hence thesis by A6,A7,A8,A9;
      end;
      assume y in F2;
      then consider i such that
A10:  y = ll.i and
A11:  1 <= i & i <= len ll and
A12:  ll.i in fixed_QC-variables(A);
      now
        assume l.i = (A)a.0;
        then ll.i = x by A6,A11,Th3;
        hence contradiction by A12,Lm1;
      end;
      then ll.i = l.i by A6,A11,Th3;
      hence y in F1 by A6,A10,A11,A12;
    end;
    then
A13: F1 = F2 by TARSKI:2;
    Fixed(P!l) = F1 & Fixed(P!ll) = F2 by QC_LANG3:64;
    hence thesis by A13,Th17;
  end;
A14: for y,p st P[p] holds P[All(y,p)]
  proof
    let y,p such that
A15: Fixed(p.x) = Fixed p;
    now
      assume x <> y;
      hence Fixed(All(y,p).x) = Fixed(All(y,p.x)) by Th25
        .= Fixed(p) by A15,QC_LANG3:68
        .= Fixed(All(y,p)) by QC_LANG3:68;
    end;
    hence thesis by Th24;
  end;
A16: P[VERUM(A)] by Th15;
  for p holds P[p] from QC_LANG1:sch 1(A5,A16,A1,A3,A14);
  hence thesis;
end;
