theorem Th43:
  m|J divides m
  proof
    let i be object;
    per cases;
    suppose
A1:   not i in I;
      dom (m|J) = I & dom m = I by PARTFUN1:def 2;
      hence thesis by A1,FUNCT_1:def 2;
    end;
    suppose
A2:   i in I;
      per cases;
      suppose i in J;
        hence thesis by A2,BAR;
      end;
      suppose not i in J;
        hence thesis by A2,BAR;
      end;
    end;
  end;
