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 Th2:
  min(min K,min N) = min(K\/N)
proof
  set m=min(min N,min K);
A1: for k be ExtReal st k in N\/K holds m<=k
  proof
    let k be ExtReal;
    assume k in N\/K;
    then k in N or k in K by XBOOLE_0:def 3;
    then
A2: min N<=k or min K <=k by XXREAL_2:def 7;
A3: m<=min K by XXREAL_0:17;
    m<=min N by XXREAL_0:17;
    hence thesis by A2,A3,XXREAL_0:2;
  end;
  m=min N or m=min K by XXREAL_0:15;
  then m in N or m in K by XXREAL_2:def 7;
  then m in N\/K by XBOOLE_0:def 3;
  hence thesis by A1,XXREAL_2:def 7;
end;
