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

theorem
  C is connected implies
 for n being Nat holds N-min L~Cage(C,n) = (Cage(C,n))/.1
proof
  assume
A1: C is connected;
  let n be Nat;
  set f = Cage(C,n);
A2: for k being Nat st 1 <= k & k+1 <= len f
  holds left_cell(f,k,Gauge(C,n)) misses C
  & right_cell(f,k,Gauge(C,n)) meets C by A1,Th31;
  f is_sequence_on Gauge(C,n) &
  ex i being Nat st 1 <= i & i+1 <= len Gauge(C,n ) &
f/.1 = Gauge(C,n)*(i,width Gauge(C,n)) & f/.2 = Gauge(C,n)*(i+1,width Gauge(C,n
)) & N-min C in cell(Gauge( C,n),i,width Gauge(C,n)-'1) & N-min C <> Gauge(C,n)
  *(i, width Gauge(C,n)-'1) by A1,Def1;
  hence thesis by A2,Th30;
end;
