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 N-most C implies ex p being Point of TOP-REAL 2 st north_halfline
  x /\ L~Cage(C,n) = {p}
proof
  set f = Cage(C,n);
  assume
A1: x in N-most C;
  then x in C by XBOOLE_0:def 4;
  then north_halfline x meets L~f by Th51;
  then consider p being object such that
A2: p in north_halfline x and
A3: p in L~f by XBOOLE_0:3;
A4: p in north_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 north_halfline x /\ L~f;
    then reconsider y = a as Point of TOP-REAL 2;
    y in north_halfline x by A5,XBOOLE_0:def 4;
    then
A6: y`1 = x`1 by TOPREAL1:def 10
      .= p`1 by A2,TOPREAL1:def 10;
    p`2 = N-bound L~f by A1,A4,Th82
      .= y`2 by A1,A5,Th82;
    then y = p by A6,TOPREAL3:6;
    hence a in {p} by TARSKI:def 1;
  end;
  thus thesis by A4,ZFMISC_1:31;
end;
