theorem Th8:
  (Partial_Intersection A1).n c= A1.n
proof
  per cases by NAT_1:6;
  suppose
    n = 0;
    hence thesis by Def1;
  end;
  suppose
    ex k being Nat st n = k+1;
    then consider k such that
A1: n = k+1;
    (Partial_Intersection A1).(k+1) = (Partial_Intersection A1).k /\ A1.(k
    +1) by Def1;
    hence thesis by A1,XBOOLE_1:17;
  end;
end;
