reserve i,k for Nat;
reserve A for QC-alphabet;
reserve x for bound_QC-variable of A;
reserve a for free_QC-variable of A;
reserve p,q for Element of QC-WFF(A);
reserve l for FinSequence of QC-variables(A);
reserve P,Q for QC-pred_symbol of A;
reserve V for non empty Subset of QC-variables(A);
reserve s,t for QC-symbol of A;

theorem Th22:
  for p,q being QC-formula of A holds p is closed & q is closed iff p
  '&' q is closed
proof
  let p,q be QC-formula of A;
  thus p is closed & q is closed implies p '&' q is closed
  proof
    assume still_not-bound_in p = {} & still_not-bound_in q = {};
    then (still_not-bound_in p) \/ (still_not-bound_in q) = {};
    hence still_not-bound_in p '&' q = {} by Th10;
  end;
  assume
A1: still_not-bound_in p '&' q = {};
  still_not-bound_in p '&' q = (still_not-bound_in p) \/ (
  still_not-bound_in q) by Th10;
  hence still_not-bound_in p={} & still_not-bound_in q={} by A1,XBOOLE_1:15;
end;
