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 Th47:
  p is conditional implies Vars(p,V) = Vars(the_antecedent_of p,V)
  \/ Vars(the_consequent_of p,V)
proof
  set p1 = the_antecedent_of p;
  set p2 = the_consequent_of p;
  assume p is conditional;
  then p = p1 => p2 by QC_LANG2:38;
  then p = 'not'(p1 '&' 'not' p2) by QC_LANG2:def 2;
  hence Vars(p,V) = Vars(p1 '&' 'not' p2, V) by Th39
    .= Vars(p1, V) \/ Vars('not' p2, V) by Th42
    .= Vars(the_antecedent_of p,V) \/ Vars(the_consequent_of p,V) by Th39;
end;
