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;
