theorem
  x in dom p & x in dom q & PP_imp(p,q).x = TRUE implies
   p.x = FALSE or q.x = TRUE
  proof
    assume that
A1: x in dom p and
A2: x in dom q and
A3: PP_imp(p,q).x = TRUE;
    p.x <> TRUE or q.x <> FALSE by A1,A2,A3,Th35;
    hence thesis by A1,A2,Th3;
  end;
