theorem Th55:
  [S,x] is quantifiable implies for a holds NEx_Val(v.((S_Bound(@
  CQCSub_All([S,x],xSQ)))|a),S,x,xSQ) = NEx_Val(v,S,x,xSQ)
proof
  assume
A1: [S,x] is quantifiable;
  set finSub = RestrictSub(x,All(x,S`1),xSQ);
  set NF1 = NEx_Val(v,S,x,xSQ);
  set S1 = CQCSub_All([S,x],xSQ);
  let a;
  set z = S_Bound(@S1);
  set NF = NEx_Val(v.(z|a),S,x,xSQ);
  v is Element of Funcs(bound_QC-variables(Al),A) by VALUAT_1:def 1;
  then ex f st v = f & dom f = bound_QC-variables(Al) & rng f c= A
   by FUNCT_2:def 2;
  then rng @finSub c= dom v;
  then
A2: dom NF1 = dom @finSub by RELAT_1:27;
  v.(z|a) is Element of Funcs(bound_QC-variables(Al),A) by VALUAT_1:def 1;
  then ex f st v.(z|a) = f & dom f = bound_QC-variables(Al) & rng f c= A by
FUNCT_2:def 2;
  then
A3: rng @finSub c= dom (v.(z|a));
  then
A4: dom NF = dom @finSub by RELAT_1:27;
  for b being object st b in dom NF holds NF.b = NF1.b
  proof
    let b being object such that
A5: b in dom NF;
A6: @finSub.b in rng @finSub by A4,A5,FUNCT_1:3;
    then reconsider x = @finSub.b as bound_QC-variable of Al;
    not z in rng finSub by A1,Th40;
    then z <> x by A6,SUBSTUT1:def 2;
    then
A7: v.(z|a).x = v.x by Th48;
    NF.b = (v.(z|a)).x by A5,FUNCT_1:12;
    hence thesis by A4,A2,A5,A7,FUNCT_1:12;
  end;
  hence thesis by A3,A2,FUNCT_1:2,RELAT_1:27;
end;
