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* n = 0
proof
  now
    per cases;
    suppose
      0=n;
      hence thesis by NAT_1:def 1;
    end;
    suppose
      0<n;
      then 0 in {l where l is Nat:l<n};
      then
A1:   0 in n by AXIOMS:4;
      n is Subset of NAT by Th8;
      hence thesis by A1,NAT_1:def 1;
    end;
  end;
  hence thesis;
end;
