reserve Al for QC-alphabet;
reserve PHI for Consistent Subset of CQC-WFF(Al),
        p,q,r,s for Element of CQC-WFF(Al),
        A for non empty set,
        J for interpretation of Al,A,
        v for Element of Valuations_in(Al,A),
        m,n,i,j,k for Nat,
        l for CQC-variable_list of k,Al,
        P for QC-pred_symbol of k,Al,
        x,y,z for bound_QC-variable of Al,
        b for QC-symbol of Al,
        PR for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve Al2 for Al-expanding QC-alphabet,
        J2 for interpretation of Al2,A,
        Jp for interpretation of Al,A,
        v2 for Element of Valuations_in(Al2,A),
        vp for Element of Valuations_in(Al,A);

theorem Th16:
 for Z being Element of CQC-Sub-WFF(Al), Z2 being Element of CQC-Sub-WFF(Al2)
 st Z`1 is universal & Z2`1 is universal & bound_in Z`1 = bound_in Z2`1 &
 the_scope_of Z`1 = the_scope_of Z2`1 & Z = Z2 holds S_Bound(@Z) = S_Bound(@Z2)
proof
  let Z be Element of CQC-Sub-WFF(Al), Z2 be Element of CQC-Sub-WFF(Al2)
   such that
A1: Z`1 is universal & Z2`1 is universal & bound_in Z`1 = bound_in Z2`1 &
    the_scope_of Z`1 = the_scope_of Z2`1 & Z = Z2;
  set p = (@Z)`1;
  set p2 = (@Z2)`1;
  set S = (@Z)`2;
  set S2 = (@Z2)`2;
  reconsider p as Element of CQC-WFF(Al) by A1,SUBSTUT1:def 35;
  reconsider p2 as Element of CQC-WFF(Al2) by A1,SUBSTUT1:def 35;
  reconsider S as CQC_Substitution of Al;
  reconsider S2 as CQC_Substitution of Al2;
  set x = bound_in p;
  set x2 = bound_in p2;
  set q = the_scope_of p;
  set q2 = the_scope_of p2;
  p is universal by A1,SUBSTUT1:def 35;
  then p = All(x,q) by QC_LANG2:6;
  then reconsider q as Element of CQC-WFF(Al) by CQC_LANG:13;
  p2 is universal by A1,SUBSTUT1:def 35;
  then p2 = All(x2,q2) by QC_LANG2:6;
  then reconsider q2 as Element of CQC-WFF(Al2) by CQC_LANG:13;
  reconsider x as bound_QC-variable of Al;
  reconsider x2 as bound_QC-variable of Al2;
A2: p =Z`1 by SUBSTUT1:def 35 .= p2 by A1,SUBSTUT1:def 35;
A3: S =Z`2 by SUBSTUT1:def 35 .= S2 by A1,SUBSTUT1:def 35;
A4: x = bound_in Z`1 by SUBSTUT1:def 35 .= x2 by A1,SUBSTUT1:def 35;
A5: q = the_scope_of Z`1 by SUBSTUT1:def 35 .= q2 by A1,SUBSTUT1:def 35;
  set rsub = RestrictSub(x,p,S);
  set rsub2 = RestrictSub(x2,p2,S2);
  per cases;
  suppose
A6: not x in rng rsub;
    then not x2 in rng rsub2 by A2,A3,A4,Th13;
    hence S_Bound @Z2 = x2 by SUBSTUT1:def 36
     .= S_Bound @Z by A4,A6,SUBSTUT1:def 36;
  end;
  suppose
A7: x in rng rsub;
    then x2 in rng rsub2 by A2,A3,A4,Th13;
    then S_Bound @Z2 = x.(upVar(rsub2,q2)) by SUBSTUT1:def 36
     .= [4,upVar(rsub2,q2)] by QC_LANG3:def 2
     .= [4,upVar(rsub,q)] by A5,A2,A3,A4,Th13,Th14
     .= x.(upVar(rsub,q)) by QC_LANG3:def 2
     .= S_Bound @Z by A7,SUBSTUT1:def 36;
    hence thesis;
  end;
end;
