theorem Th9:
  A1.n c= (Partial_Union A1).n
proof
  per cases by NAT_1:6;
  suppose
    n = 0;
    hence thesis by Def2;
  end;
  suppose
    ex k being Nat st n = k+1;
    then consider k such that
A1: n = k+1;
    (Partial_Union A1).(k+1) = (Partial_Union A1).k \/ A1.(k+1) by Def2;
    hence thesis by A1,XBOOLE_1:7;
  end;
end;
