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 Th53:
  K is having_valuation implies
  for S being Subset of ValuatRing v holds 0 is LowerBound of v.:S
  proof
    assume
A1: K is having_valuation;
    let S be Subset of ValuatRing v;
    let x be ExtReal;
    assume x in v.:S;
    then
A2: ex c being Element of K st c in S & x = v.c by FUNCT_2:65;
    the carrier of ValuatRing v = NonNegElements v by A1,Def12;
    hence thesis by A2,Th47;
  end;
