reserve x for object;
reserve D for set;
reserve p for PartialPredicate of D;
reserve D for non empty set;
reserve p,q,r for PartialPredicate of D;

theorem Th12:
  x in dom q & PP_or(p,q).x = FALSE implies q.x = FALSE
  proof
    assume that
A1: x in dom q and
A2: PP_or(p,q).x = FALSE;
    q.x <> TRUE by A1,A2,Def4;
    hence thesis by A1,Th3;
  end;
