reserve Al for QC-alphabet;
reserve a,b,b1 for object,
  i,j,k,n for Nat,
  p,q,r,s for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  P for QC-pred_symbol of k,Al,
  l,ll for CQC-variable_list of k,Al,
  Sub,Sub1 for CQC_Substitution of Al,
  S,S1,S2 for Element of CQC-Sub-WFF(Al),
  P1,P2 for Element of QC-pred_symbols(Al);

theorem Th11:
  (for Sub holds ex S st S`1 = p & S`2 = Sub) implies for Sub
  holds ex S st S`1 = All(x,p) & S`2 = Sub
proof
  assume
A1: for Sub holds ex S st S`1 = p & S`2 = Sub;
  let Sub;
  set Sub1 = @RestrictSub(x,All(x,p),Sub) +* (x|S_Bound([All(x,p),Sub]));
  Sub1 is CQC_Substitution of Al iff Sub1 is Element of
    PFuncs(bound_QC-variables(Al),bound_QC-variables(Al)) by SUBSTUT1:def 1;
  then reconsider Sub1 as CQC_Substitution of Al
   by PARTFUN1:45;
  ex S st S`1 = p & S`2 = Sub1 by A1;
  then consider S1 such that
A2: S1 = [All(x,p),Sub] by Th10;
  take S1;
  thus thesis by A2;
end;
