reserve X,x,y,z for set;
reserve n,m,k,k9,d9 for Nat;
reserve d for non zero Nat;
reserve i,i0,i1 for Element of Seg d;
reserve l,r,l9,r9,l99,r99,x,x9,l1,r1,l2,r2 for Element of REAL d;
reserve Gi for non trivial finite Subset of REAL;
reserve li,ri,li9,ri9,xi,xi9 for Real;
reserve G for Grating of d;

theorem Th9:
  for X being non empty finite Subset of REAL holds
  ex ri being Element of REAL st ri in X & for xi st xi in X holds ri >= xi
proof
  defpred P[set] means
   ex ri being Element of REAL st ri in $1 &
   for xi st xi in $1 holds ri >= xi;
  let X be non empty finite Subset of REAL;
A1: for xi being Element of REAL st xi in X holds P[{xi}]
  proof
    let xi be Element of REAL;
    assume xi in X;
    take xi;
    thus thesis by TARSKI:def 1;
  end;
A2: for x being Element of REAL, B being non empty finite Subset of REAL st
  x in X & B c= X & not x in B & P[B] holds P[B \/ {x}]
  proof
    let x be Element of REAL;
    let B be non empty finite Subset of REAL;
    assume that x in X and B c= X
    and not x in B and
A3: P[B];
    consider ri such that
A4: ri in B and
A5: for xi st xi in B holds ri >= xi by A3;
    set B9 = B \/ {x};
A6: now
      let xi;
      xi in {x} iff xi = x by TARSKI:def 1;
      hence xi in B9 iff xi in B or xi = x by XBOOLE_0:def 3;
    end;
    per cases;
    suppose
A7:   x <= ri;
       reconsider ri as Element of REAL by XREAL_0:def 1;
      take ri;
      thus ri in B9 by A4,A6;
      let xi;
      assume xi in B9;
      then xi in B or xi = x by A6;
      hence thesis by A5,A7;
    end;
    suppose
A8:   ri < x;
      take x;
      thus x in B9 by A6;
      let xi;
      assume xi in B9;
      then xi in B or xi = x by A6;
      then ri >= xi or xi = x by A5;
      hence thesis by A8,XXREAL_0:2;
    end;
  end;
  thus P[X] from NonEmptyFinite(A1,A2);
end;
