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 Th53:
  p in C implies south_halfline p meets L~Cage(C,n)
proof
  set f = Cage(C,n);
  assume
A1: p in C;
  set X = {q where q is Point of TOP-REAL 2: q`1 = p`1 & q`2 <= p`2};
A2: X = south_halfline p by TOPREAL1:34;
  min(S-bound (L~f),p`2)-1 < S-bound (L~f)-0 by XREAL_1:15,XXREAL_0:17;
  then
  f/.1 = N-min L~f & |[p`1,min(S-bound (L~f),p`2)-1]|`2 < S-bound (L~f) by
EUCLID:52,JORDAN9:32;
  then |[p`1,min(S-bound (L~f),p`2)-1]| in LeftComp f by JORDAN2C:112;
  then
A3: |[p`1,min(S-bound (L~f),p`2)-1]| in UBD L~f by GOBRD14:36;
  LeftComp f is_outside_component_of L~f by GOBRD14:34;
  then LeftComp f is_a_component_of (L~f)` by JORDAN2C:def 3;
  then
A4: UBD L~f is_a_component_of (L~f)` by GOBRD14:36;
  reconsider X as connected Subset of TOP-REAL 2 by A2;
A5: C c= BDD L~f & p in X by JORDAN10:12;
  min(S-bound (L~f),p`2) <= p`2 by XXREAL_0:17;
  then min(S-bound (L~f),p`2)-1 <= p`2-0 by XREAL_1:13;
  then
A6: |[p`1,min(S-bound (L~f),p`2)-1]|`2 <= p`2 by EUCLID:52;
  |[p`1,min(S-bound (L~f),p`2)-1]|`1 = p`1 by EUCLID:52;
  then |[p`1,min(S-bound (L~f),p`2)-1]| in X by A6;
  then
A7: X meets UBD L~f by A3,XBOOLE_0:3;
  assume not thesis;
  then X c= (L~f)` by A2,SUBSET_1:23;
  then X c= UBD L~f by A7,A4,GOBOARD9:4;
  then p in BDD L~f /\ UBD L~f by A1,A5,XBOOLE_0:def 4;
  then BDD L~f meets UBD L~f;
  hence contradiction by JORDAN2C:24;
end;
