theorem Th46:
  still_not-bound_in p c= Bound_Vars(p) implies still_not-bound_in
  All(x,p) c= Bound_Vars(All(x,p))
proof
A1: still_not-bound_in All(x,p) = (still_not-bound_in p) \ {x} by QC_LANG3:12;
  All(x,p) is universal by QC_LANG1:def 21;
  then Bound_Vars(All(x,p)) = Bound_Vars(the_scope_of All(x,p)) \/ {bound_in
  All(x,p)} by SUBSTUT1:6;
  then Bound_Vars(All(x,p)) = Bound_Vars(p) \/ {bound_in All(x,p)} by
QC_LANG2:7;
  then
  Bound_Vars(p) \ {x} c= Bound_Vars(p) & Bound_Vars(p) c= Bound_Vars(All(x
  ,p)) by XBOOLE_1:7,36;
  then
A2: Bound_Vars(p) \ {x} c= Bound_Vars(All(x,p));
  assume still_not-bound_in p c= Bound_Vars(p);
  then still_not-bound_in All(x,p) c= Bound_Vars(p) \ {x} by A1,XBOOLE_1:33;
  hence thesis by A2;
end;
