reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;

theorem
  min(min* Ke,min* Ne) <= min* (Ke\/Ne)
proof
  now
    per cases;
    suppose
      Ke is empty or Ne is empty;
      then min* Ke=0 & min* Ne >=0 or min* Ne=0 & min*Ke>=0 by NAT_1:def 1;
      hence thesis by XXREAL_0:def 9;
    end;
    suppose
      Ke is non empty & Ne is non empty;
      then reconsider K=Ke,N=Ne as non empty Subset of NAT;
A1:   min N=min* Ne by Th1;
A2:   min(K\/N)=min*(Ke\/Ne) by Th1;
      min K=min* Ke by Th1;
      hence thesis by A1,A2,Th2;
    end;
  end;
  hence thesis;
end;
