theorem
  m - {$}P = m
proof
A1: now
    let p;
    assume p in P;
    then ({$}P).p = 0 by Lm1;
    hence m.p = m.p - ({$}P).p;
  end;
  {$}P c= m by Th1;
  hence thesis by A1,Def5;
end;
