reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem
  x in W-most C implies ex p being Point of TOP-REAL 2 st west_halfline
  x /\ L~Cage(C,n) = {p}
proof
  set f = Cage(C,n);
  assume
A1: x in W-most C;
  then x in C by XBOOLE_0:def 4;
  then west_halfline x meets L~f by Th54;
  then consider p being object such that
A2: p in west_halfline x and
A3: p in L~f by XBOOLE_0:3;
A4: p in west_halfline x /\ L~f by A2,A3,XBOOLE_0:def 4;
  reconsider p as Point of TOP-REAL 2 by A2;
  take p;
  hereby
    let a be object;
    assume
A5: a in west_halfline x /\ L~f;
    then reconsider y = a as Point of TOP-REAL 2;
    y in west_halfline x by A5,XBOOLE_0:def 4;
    then
A6: y`2 = x`2 by TOPREAL1:def 13
      .= p`2 by A2,TOPREAL1:def 13;
    p`1 = W-bound L~f by A1,A4,Th85
      .= y`1 by A1,A5,Th85;
    then y = p by A6,TOPREAL3:6;
    hence a in {p} by TARSKI:def 1;
  end;
  thus thesis by A4,ZFMISC_1:31;
end;
