reserve A for QC-alphabet;
reserve i,j,k,l,m,n for Nat;
reserve a,b,e for set;
reserve t,u,v,w,z for QC-symbol of A;
reserve p,q,r,s for Element of CQC-WFF(A);
reserve x for Element of bound_QC-variables(A);
reserve ll for CQC-variable_list of k,A;
reserve P for QC-pred_symbol of k,A;
reserve f,h for Element of Funcs(bound_QC-variables(A),bound_QC-variables(A)),
  K,L for Element of Fin bound_QC-variables(A);

theorem Th44:
  still_not-bound_in p = still_not-bound_in SepVar p
proof
  defpred P[Element of CQC-WFF(A)] means for t,K,f st [$1,t,K,f]
  in SepQuadruples p holds f.:(still_not-bound_in $1) = still_not-bound_in
  (((SepFunc(A)).$1 qua Element of Funcs([:QC-symbols(A),
  Funcs(bound_QC-variables(A), bound_QC-variables(A)):],
  CQC-WFF(A))).([t,f] qua Element of [:QC-symbols(A),
  Funcs(bound_QC-variables(A),bound_QC-variables(A)):]) qua Element of
  CQC-WFF(A));
A1: [p,index p,{}.bound_QC-variables(A),id bound_QC-variables(A)] in
  SepQuadruples p by Th30;
A2: now
    let r;
    reconsider g = (SepFunc(A)).r as Function of [:QC-symbols(A),
     Funcs(bound_QC-variables(A), bound_QC-variables(A)):], CQC-WFF(A);
    assume
A3: P[r];
A4: (SepFunc(A)).('not' r) = NEGATIVE(g) by Def7;
    thus P['not' r]
    proof
      let u,K,f;
      assume ['not' r,u,K,f] in SepQuadruples p;
      then
A5:   [r,u,K,f] in SepQuadruples p by Th31;
      set uf = [u,f];
      reconsider r9 = g.uf as Element of CQC-WFF(A);
A6:   still_not-bound_in r9 = still_not-bound_in 'not' r9 by QC_LANG3:7;
A7:   still_not-bound_in r = still_not-bound_in 'not' r by QC_LANG3:7;
      (NEGATIVE g).uf = 'not' r9 by Def2;
      hence thesis by A4,A3,A7,A6,A5;
    end;
  end;
A8: now
    let k;
    let l be CQC-variable_list of k,A;
    let P be QC-pred_symbol of k,A;
    thus P[P!l]
    proof
      let u,K,f such that
      [P!l,u,K,f] in SepQuadruples p;
      set fl = f*l;
A9:   f.:{ l.i : 1 <= i & i <= len l & l.i in bound_QC-variables(A) } = { (
      fl).j : 1 <= j & j <= len fl & fl.j in bound_QC-variables(A) }
      proof
A10:    len fl = k by CARD_1:def 7
          .= len l by CARD_1:def 7;
        thus f.:{ l.i : 1 <= i & i <= len l & l.i in bound_QC-variables(A) } c=
         {fl.j : 1 <= j & j <= len fl & fl.j in bound_QC-variables(A) }
        proof
          let x be object;
          assume
          x in f.: { l.i : 1 <= i & i <= len l & l.i in bound_QC-variables(A)};
          then consider y being object such that
A11:      y in dom f & y in { l.i : 1 <= i & i <= len l & l.i in
          bound_QC-variables(A) } & x = f.y by FUNCT_1:def 6;
          consider i such that
A12:      y = l.i and
A13:      1 <= i and
A14:      i <= len l and
          l.i in bound_QC-variables(A) by A11;
          i in dom l by A13,A14,FINSEQ_3:25;
          then
A15:      f.(l.i) = fl.i by FUNCT_1:13;
          fl.i in bound_QC-variables(A) by A10,A13,A14,Th13;
          hence thesis by A10,A11,A12,A13,A14,A15;
        end;
        let x be object;
        assume x in {fl.i: 1<=i & i<=len fl & fl.i in bound_QC-variables(A) };
        then consider i such that
A16:    x = fl.i and
A17:    1 <= i and
A18:    i <= len fl and
        fl.i in bound_QC-variables(A);
        i in dom l by A10,A17,A18,FINSEQ_3:25;
        then
A19:    fl.i = f.(l.i) by FUNCT_1:13;
A20:    l.i in bound_QC-variables(A) by A10,A17,A18,Th13;
        then
A21:    l.i in dom f by FUNCT_2:def 1;
        l.i in { l.j : 1 <= j & j <= len l & l.j in bound_QC-variables(A) }
        by A10,A17,A18,A20;
        hence thesis by A16,A21,A19,FUNCT_1:def 6;
      end;
A22:  f.:still_not-bound_in (P!l) = f.:still_not-bound_in l by QC_LANG3:5
        .= f.:variables_in(l,bound_QC-variables(A)) by QC_LANG3:2
        .= variables_in(fl,bound_QC-variables(A)) by A9
        .= still_not-bound_in fl by QC_LANG3:2
        .= still_not-bound_in (P!fl) by QC_LANG3:5;
      ATOMIC(P,l).(u,f) = P!(f*l) by Def5;
      hence thesis by A22,Def7;
    end;
  end;
A23: now
    let r,x such that
A24: P[r];
    thus P[All(x, r)]
    proof
      reconsider g = (SepFunc(A)).r as Function of
       [:QC-symbols(A),Funcs(bound_QC-variables(A),bound_QC-variables(A)):],
       CQC-WFF(A);
      let u,K,f such that
A25:  [All(x,r),u,K,f] in SepQuadruples p;
A26:  [r,u++,K \/ {.x .}, f+*(x .--> x.u)] in SepQuadruples p by A25,Th33;
      f+*(x .--> x.u) is Function of bound_QC-variables(A),
      bound_QC-variables(A) by Lm1;
      then reconsider
      fu = f +* (x .--> x.u) as Element of Funcs(bound_QC-variables(A)
      ,bound_QC-variables(A)) by FUNCT_2:8;
      reconsider r99 = g.(u++,fu) as Element of CQC-WFF(A);
A27:  UNIVERSAL(x,g).(u,f) = All(x.u,r99) by Def4;
A28:  still_not-bound_in All(x, r) = still_not-bound_in r \ {x} by QC_LANG3:12;
      then
A29:  not x.u in f.:(still_not-bound_in r \ {x}) by A25,Th43;
      thus f.:(still_not-bound_in All(x, r)) = fu.:(still_not-bound_in r \ {x}
      ) by A28,Th3
       .= fu.:(still_not-bound_in r) \ {x.u} by A29,Th4
       .= still_not-bound_in r99 \ {x.u} by A24,A26
       .= still_not-bound_in All(x.u,r99) by QC_LANG3:12
       .= still_not-bound_in (((SepFunc(A)).(All(x, r)) qua Element of Funcs([:
      QC-symbols(A),Funcs(bound_QC-variables(A),bound_QC-variables(A)):],
      CQC-WFF(A))).([u,f] qua Element of [:QC-symbols(A),
      Funcs(bound_QC-variables(A), bound_QC-variables(A)):])
      qua Element of CQC-WFF(A)) by A27,Def7;
    end;
  end;
A30: now
    let r,s such that
A31: P[r] and
A32: P[s];
    thus P[r '&' s]
    proof
      reconsider g = (SepFunc(A)).r, h = (SepFunc(A)).s as Function of
       [:QC-symbols(A),Funcs(bound_QC-variables(A),bound_QC-variables(A)):],
       CQC-WFF(A);
      let u,K,f such that
A33:  [r '&' s, u, K, f] in SepQuadruples p;
      reconsider r9=g.(u,f), s9 = h.(u+QuantNbr(r),f) as Element of CQC-WFF(A);
A34:  CON(g,h,QuantNbr(r)).(u,f) = r9 '&' s9 by Def3;
      [r,u,K,f] in SepQuadruples p by A33,Th32;
      then
A35:  f.:(still_not-bound_in r) = still_not-bound_in r9 by A31;
      [s,u+QuantNbr(r),K,f] in SepQuadruples p by A33,Th32;
      then
A36:  f.:(still_not-bound_in s) = still_not-bound_in s9 by A32;
      thus f.:(still_not-bound_in r '&' s) = f.:(still_not-bound_in r \/
      still_not-bound_in s) by QC_LANG3:10
      .= still_not-bound_in r9 \/ still_not-bound_in s9 by A35,A36,RELAT_1:120
      .= still_not-bound_in(r9 '&' s9) by QC_LANG3:10
      .= still_not-bound_in (((SepFunc(A)).(r '&' s) qua Element of
    Funcs([:QC-symbols(A),Funcs(bound_QC-variables(A),bound_QC-variables(A)):],
    CQC-WFF(A))).([u,f] qua Element of [:QC-symbols(A),
    Funcs(bound_QC-variables(A),bound_QC-variables(A)):]) qua Element of
    CQC-WFF(A)) by A34,Def7;
    end;
  end;
A37: (SepFunc(A)).VERUM(A) = [:QC-symbols(A),Funcs(bound_QC-variables(A),
  bound_QC-variables(A)):] --> VERUM(A) by Def7;
A38: P[VERUM(A)]
  proof
    let v,K,f such that
    [VERUM(A),v,K,f] in SepQuadruples p;
A39: still_not-bound_in VERUM(A) = {} by QC_LANG3:3;
    thus thesis by A39,A37;
  end;
A40: for q holds P[q] from CQCInd(A38,A8,A2,A30,A23);
  thus still_not-bound_in p =
   (id bound_QC-variables(A)).:(still_not-bound_in p)
  by FUNCT_1:92
    .= still_not-bound_in SepVar p by A40,A1;
end;
