reserve x, y, z, s for ExtReal;
reserve i, j for Integer;
reserve n, m for Nat;
reserve x, y, v, u for ExtInt;
reserve
  D for non empty doubleLoopStr,
  A for Subset of D;
reserve K for Field-like non degenerated
  associative add-associative right_zeroed right_complementable
  distributive Abelian non empty doubleLoopStr,
  a, b, c for Element of K;
reserve v for Valuation of K;

theorem Th64:
  for S being non empty Subset of K st
  K is having_valuation & S is Subset of ValuatRing v
  holds min(S,v) c= S
  proof
    let S be non empty Subset of K;
    assume K is having_valuation & S is Subset of ValuatRing v; then
    min(S,v) = v"{inf(v.:S)} /\ S by Def14;
    hence min(S,v) c= S by XBOOLE_1:17;
  end;
