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 Th10:
  S`2 = @RestrictSub(x,All(x,p),Sub) +* (x|S_Bound([All(x,p),Sub])
  ) & S`1 = p implies [S,x] is quantifiable & ex S1 st S1 = [All(x,p),Sub]
proof
  set Sub1 = @RestrictSub(x,All(x,p),Sub) +* (x|S_Bound([All(x,p),Sub]));
  reconsider Sub as CQC_Substitution of Al;
  assume that
A1: S`2 = Sub1 and
A2: S`1 = p;
A3: [S,x]`2 = x & ([S,x]`1)`1 = p by A2;
A4: the_scope_of All(x,p) = p & All(x,p) is universal by QC_LANG1:def 21
,QC_LANG2:7;
  Sub1 = ExpandSub(x,p,RestrictSub(x,All(x,p),Sub)) & bound_in All(x,p) =
  x by Th9,QC_LANG2:7;
  then All(x,p),Sub PQSub Sub1 by A4,SUBSTUT1:def 14;
  then consider a such that
A5: a = [[All(x,p),Sub],Sub1] and
A6: All(x,p),Sub PQSub Sub1;
  a in QSub(Al) by A5,A6,SUBSTUT1:def 15;
  then
A7: (QSub(Al)).[All(x,p),Sub] = Sub1 by A5,FUNCT_1:1;
A8: ([S,x]`1)`2 = Sub1 by A1;
  hence [S,x] is quantifiable by A7,A3,SUBSTUT1:def 22;
A9: [S,x] is quantifiable by A7,A8,A3,SUBSTUT1:def 22;
  then reconsider Sub as second_Q_comp of [S,x] by A7,A8,A3,SUBSTUT1:def 23;
  take S1 = CQCSub_All([S,x],Sub);
  S1 = Sub_All([S,x],Sub) by A9,SUBLEMMA:def 5;
  hence thesis by A3,A9,SUBSTUT1:def 24;
end;
