reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;

theorem
  for C being Subset of TOP-REAL 2 holds {p where p is Point of TOP-REAL
  2: p`1 < W-bound C} is non empty convex connected Subset of TOP-REAL 2
proof
  let C be Subset of TOP-REAL 2;
  set A = {p where p is Point of TOP-REAL 2: p`1 < W-bound C};
  A c= the carrier of TOP-REAL 2
  proof
    let a be object;
    assume a in A;
    then ex p being Point of TOP-REAL 2 st a = p & p`1 < W-bound C;
    hence thesis;
  end;
  then reconsider A as Subset of TOP-REAL 2;
  set p = W-bound C;
  set b = |[p-1,0]|;
A1: b`1 = p-1;
  p - 1 < p - 0 by XREAL_1:15;
  then
A2: b in A by A1;
  A is convex
  proof
    let w1, w2 be Point of TOP-REAL 2;
    assume w1 in A;
    then consider p being Point of TOP-REAL 2 such that
A3: w1 = p and
A4: p`1 < W-bound C;
    assume w2 in A;
    then consider q being Point of TOP-REAL 2 such that
A5: w2 = q and
A6: q`1 < W-bound C;
    let k be object;
    assume
A7: k in LSeg(w1,w2);
    then reconsider z = k as Point of TOP-REAL 2;
    per cases by XXREAL_0:1;
    suppose
      p`1 < q`1;
      then z`1 <= w2`1 by A3,A5,A7,TOPREAL1:3;
      then z`1 < W-bound C by A5,A6,XXREAL_0:2;
      hence thesis;
    end;
    suppose
      q`1 < p`1;
      then z`1 <= w1`1 by A3,A5,A7,TOPREAL1:3;
      then z`1 < W-bound C by A3,A4,XXREAL_0:2;
      hence thesis;
    end;
    suppose
      p`1 = q`1;
      then z`1 = p`1 by A3,A5,A7,GOBOARD7:5;
      hence thesis by A4;
    end;
  end;
  then reconsider A as non empty convex Subset of TOP-REAL 2 by A2;
  A is connected;
  hence thesis;
end;
