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
  not min* Ne in Ne/\Ke implies min* Ne = min* (Ne\Ke)
proof
  assume
A1: not min* Ne in Ne/\Ke;
  now
    per cases;
    suppose
      Ne is empty;
      hence thesis;
    end;
    suppose
      Ne is non empty;
      then
A2:   min* Ne in Ne by NAT_1:def 1;
      then not min* Ne in Ke by A1,XBOOLE_0:def 4;
      then
A3:   min* Ne in Ne\Ke by A2,XBOOLE_0:def 5;
      then
A4:   min* (Ne\Ke)<=min* Ne by NAT_1:def 1;
A5:   Ne\Ke c=Ne by XBOOLE_1:36;
      min* (Ne\Ke) in Ne\Ke by A3,NAT_1:def 1;
      then min* Ne<=min* (Ne\Ke) by A5,NAT_1:def 1;
      hence thesis by A4,XXREAL_0:1;
    end;
  end;
  hence thesis;
end;
