
theorem Th29:
  for G be Go-board for f be FinSequence of TOP-REAL 2 for i,j be
  Nat holds f is_sequence_on G & f is special & i <= len G & j <=
  width G implies cell(G,i,j)\L~f is connected
proof
  let G be Go-board;
  let f be FinSequence of TOP-REAL 2;
  let i,j be Nat;
  assume that
A1: f is_sequence_on G and
A2: f is special and
A3: i<=len G and
A4: j<=width G;
  Int cell(G,i,j) misses L~f by A1,A2,A3,A4,JORDAN9:14;
  then
A5: Int cell(G,i,j) c= (L~f)` by SUBSET_1:23;
  cell(G,i,j)\L~f c= cell(G,i,j) by XBOOLE_1:36;
  then
A6: cell(G,i,j)\L~f c= Cl Int cell(G,i,j) by A3,A4,GOBRD11:35;
A7: Int cell(G,i,j) c= cell(G,i,j) by TOPS_1:16;
A8: Int cell(G,i,j) is convex by A3,A4,GOBOARD9:17;
  cell(G,i,j)\L~f = cell(G,i,j) /\ (L~f)` by SUBSET_1:13;
  then Int cell(G,i,j) c= cell(G,i,j)\L~f by A5,A7,XBOOLE_1:19;
  hence thesis by A6,A8,CONNSP_1:18;
end;
