theorem Th45:
  still_not-bound_in p c= Bound_Vars(p) & still_not-bound_in q c=
  Bound_Vars(q) implies still_not-bound_in (p '&' q) c= Bound_Vars(p '&' q)
proof
A1: still_not-bound_in (p '&' q) = still_not-bound_in p \/
  still_not-bound_in q by QC_LANG3:10;
  p '&' q is conjunctive by QC_LANG1:def 20;
  then Bound_Vars(p '&' q) = Bound_Vars(the_left_argument_of (p '&' q)) \/
  Bound_Vars(the_right_argument_of (p '&' q)) by SUBSTUT1:5;
  then
  Bound_Vars(p '&' q) = Bound_Vars(p) \/ Bound_Vars(the_right_argument_of
  (p '&' q)) by QC_LANG2:4;
  then
A2: Bound_Vars(p '&' q) = Bound_Vars(p) \/ Bound_Vars(q) by QC_LANG2:4;
  assume still_not-bound_in p c= Bound_Vars(p) & still_not-bound_in q c=
  Bound_Vars(q );
  hence thesis by A2,A1,XBOOLE_1:13;
end;
