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 Th9:
  ExpandSub(x,p,RestrictSub(x,All(x,p),Sub)) = @RestrictSub(x,All(x
  ,p),Sub) +* (x|S_Bound([All(x,p),Sub]))
proof
  set finSub = RestrictSub(x,All(x,p),Sub);
A1: now
    reconsider F = {[x,x.upVar(finSub,p)]} as Function;
    dom F = {x} by RELAT_1:9;
    then dom finSub misses dom F by Th6;
    then dom @finSub misses dom F by SUBSTUT1:def 2;
    then
A2: @finSub \/ F = @finSub +* F by FUNCT_4:31;
    assume
A3: x in rng finSub;
    then ExpandSub(x,p,finSub) = finSub \/ F by SUBSTUT1:def 13;
    then {[x,x.upVar(finSub,p)]} = x .--> x.upVar(finSub,p) &
    ExpandSub(x,p,finSub) = @finSub +* F by A2,FUNCT_4:82,SUBSTUT1:def 2;
    hence thesis by A3,Th7;
  end;
  now
    reconsider F = {[x,x]} as Function;
    dom F = {x} by RELAT_1:9;
    then dom finSub misses dom F by Th6;
    then dom @finSub misses dom F by SUBSTUT1:def 2;
    then
A4: @finSub \/ F = @finSub +* F by FUNCT_4:31;
    assume
A5: not x in rng finSub;
    then ExpandSub(x,p,finSub) = finSub \/ F by SUBSTUT1:def 13;
    then {[x,x]} = x .--> x & ExpandSub(x,p,finSub) = @finSub +* F by A4,
FUNCT_4:82,SUBSTUT1:def 2;
    hence thesis by A5,Th8;
  end;
  hence thesis by A1;
end;
