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 Th65:
  for S being non empty Subset of K st
  K is having_valuation & S is Subset of ValuatRing v
  for x being Element of K holds x in min(S,v) iff x in S &
  for y being Element of K st y in S holds v.x <= v.y
  proof
    let S be non empty Subset of K;
    assume
A1: K is having_valuation;
    assume
A2: S is Subset of ValuatRing v;
A3: min(S,v) = v"{inf(v.:S)} /\ S by A1,A2,Def14;
A4: inf(v.:S) is LowerBound of v.:S by XXREAL_2:def 4;
    let x be Element of K;
    hereby
      assume
A5:   x in min(S,v); then
      x in v"{inf(v.:S)} by A3,XBOOLE_0:def 4;
      then v.x in {inf(v.:S)} by FUNCT_2:38;
      then
A6:   v.x = inf(v.:S) by TARSKI:def 1;
      min(S,v) c= S by A1,A2,Th64;
      hence x in S by A5;
      let y be Element of K;
      assume y in S;
      hence v.x <= v.y by A6,A4,FUNCT_2:35,XXREAL_2:def 2;
    end;
    assume that
A7: x in S and
A8: for y being Element of K st y in S holds v.x <= v.y;
A9: v.x is LowerBound of v.:S
    proof
      let a be ExtReal;
      assume a in v.:S; then
      ex y being Element of K st y in S & a = v.y by FUNCT_2:65;
      hence v.x <= a by A8;
    end;
    for y being LowerBound of v.:S holds y <= v.x
    by A7,FUNCT_2:35,XXREAL_2:def 2;
    then v.x = inf(v.:S) by A9,XXREAL_2:def 4; then
    v.x in {inf(v.:S)} by TARSKI:def 1; then
    x in v"{inf(v.:S)} by FUNCT_2:38;
    hence x in min(S,v) by A7,A3;
  end;
