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 Th49:
  p is biconditional implies Vars(p,V) = Vars(the_left_side_of p,V
  ) \/ Vars(the_right_side_of p,V)
proof
  set p1 = the_left_side_of p;
  set p2 = the_right_side_of p;
  assume p is biconditional;
  then p = p1 <=> p2 by QC_LANG2:39;
  then p = (p1 => p2) '&' (p2 => p1) by QC_LANG2:def 4;
  hence Vars(p,V) = Vars(p1 => p2, V) \/ Vars(p2 => p1, V) by Th42
    .= Vars(p1,V) \/ Vars(p2,V) \/ Vars(p2 => p1, V) by Th48
    .= Vars(p1,V) \/ Vars(p2,V) \/ (Vars(p1,V) \/ Vars(p2,V)) by Th48
    .= Vars(the_left_side_of p,V) \/ Vars(the_right_side_of p,V);
end;
